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FUNCTIONS OF A COMPLEX VAKIABLE 



MACMILLAN AND CO., LIMITED 

LONDON BOMBAY CALCUTTA MADRAS 
MELBOURNE 

THE MACMILLAN COMPANY 

NEW YORK BOSTON CHICAGO 
DALLAS SAN FRANCISCO 

THE MACMILLAN CO. OF CANADA, LTD. 

TORONTO 



FUNCTIONS 

OF A 

COMPLEX VARIABLE 



BY 

THOMAS M. MACROBERT 

M.A., B.Sc. , 

LECTURER IN MATHEMATICS IN THE UNIVERSITY OF GLASGOW 
FORMERLY SCHOLAR OF TRINITY COLLEGE, CAMBRIDGE 




MACMILLAN AND CO., LIMITED 

ST. MARTIN'S STREET, LONDON 

1917 



COPYRIGHT 



GLASGOW: PRINTED AT THE UNIVERSITY PRESS 

BY ROBERT MACLEHOSE AND CO. LTD. 



PEEFACE 

THIS book is designed for students who, having acquired a good 
working knowledge of the calculus, desire to become acquainted 
with the theory of functions of a complex variable, and with the 
principal applications of that theory. In order to avoid making 
the subject too difficult for beginners, I have abstained from the 
use of strictly arithmetical methods, and have, while endeavour- 
ing to make the proofs sufficiently rigorous, based them mainly 
on geometrical conceptions. 

The first two chapters are intended to familiarise the student 
with the geometrical representation of complex numbers and 
of the simpler rational and irrational functions of a complex 
variable. 

In Chapter III. the properties of holomorphic functions 
are established; these properties are then used to define the 
Exponential, Circular, Logarithmic, and other transcendental func- 
tions for the domain of the complex variable, their properties as 
functions of a real variable being assumed to be known. It is 
thus possible in Chapter IV. to make use of these functions in 
examples on integration ; such examples are both interesting and 
important, and it seems desirable to introduce them to the student 
in a manner that does not involve the difficulties of complex 
series. As a preliminary to Green's Theorem I have given a 
short account of curvilinear integrals. Two proofs of Cauchy's 
Theorem are given, only the first of which depends on Green's 
Theorem. A large number of examples on contour integration 
are worked out, and here, as throughout the book, the text is 
plentifully illustrated by diagrams. 



vi PREFACE 

In view of the very full exposition of the subject given by 
Dr. Bromwich, it has been thought unnecessary to give a de- 
tailed account of infinite series. A summary of those theorems 
which are used in the book will be found at the beginning of 
Chapter V. ; the theory of uniform convergence is dealt with in 
Chapter VI. 

The remaining chapters are devoted to the applications of the 
subject. Chapter VII. includes, among other matters, the theory 
of Analytical Continuation ; various examples of the applications 
of that theory are given there and in Chapters VIII. and XV. 
The asymptotic expansions of the Gamma Function in Chapter 
VIII. and of the Bessel Functions in Chapter XV. are worked 
out for complex values of the variable. 

Chapters IX. to XI. deal with Elliptic Integrals and Functions. 
In Chapter IX. the student is shown how to reduce and evaluate 
elliptic integrals. In Chapter XI. I have established the exist- 
ence of the Jacobian Functions by considering the values of the 
Weierstrassian Function when one period is real and the other is 
purely imaginary. 

The last four chapters of the book contain a discussion of the 
theory of linear differential equations. As the most important 
of these equations are of the second order, it has been thought 
unnecessary to consider equations of higher order than the second. 
The Hypergeometric Function and Spherical and Cylindrical 
Harmonics are discussed as they arise through the solution of 
their differential equations; other properties of these functions 
are given in examples, with, in most cases, hints as to the methods 
of solution. No attempt has been made to deal with the applica- 
tions of these functions to physics, but it is hoped that the 
applied mathematician will find in these pages ready access to 
the instruments which he requires. 

Numerous examples have been given throughout the book, 
and there is also a set of Miscellaneous Examples, arranged to 
correspond with the order of the text. 

The writing of the book was undertaken at the suggestion 
of Professor George A. Gibson, LL.D., to whom I have been 



PREFACE vii 

indebted for important criticisms at all stages of the work. I 
have also to thank my colleagues, Mr. Robert J. T. Bell, D.Sc., 
and Mr. Arthur S. Morrison, M.A., B.Sc., for their assistance in 
correcting the proofs. 

Acknowledgment has been made, in foot-notes to the text, of 
various sources from which I have derived assistance. Of the 
books which I have found helpful I would particularly name 
Lindelof s Calcul des Residus, Cauchy's Mtfmoire sur les inte- 
grates definies, Jordan's GOUTS d' Analyse, and Forsyth's Theory 
of Differential Equations. I have also made use of lectures by 
Mr. R. A. Herman, M.A., and Professor E. W. Hobson, Sc.D. 

In conclusion, I would express my thanks to Messrs. MacLehose 
for the excellence of their printing work. 

THOMAS M. MACROBERT. 

GLASGOW, September 1916. 



CONTENTS 
CHAPTER I. 

COMPLEX NUMBERS. 

SECT. PAGE 

1. Definition of Complex Numbers 1 

2. Geometrical Representation of Complex Numbers 1 

3. Modulus and Amplitude. Vectors - 2 

4. Geometrical Representation of Addition. Subtraction- - - 3 

5. Multiplication and Division 4 

6. Root Extraction 5 

Examples I. - 6 

CHAPTER II. 
FUNCTIONS OF A COMPLEX VARIABLE. 

7. Uniform and Multiple- valued Functions. Path of Variation - 7 

8. Transformations. Point at Infinity - ... 7 

9. Geometrical Representation of Functions. Branch Points - 10 

10. Roots of Equations - - - - - 16 

Examples II. ----- 19 

CHAPTER III. 
HOLOMORPHIC FUNCTIONS. 

11. Limits - 22 

12. Continuity . . 3 

13. Uniform Continuity. Functions of Two Real Variables - - 24 

14. Differentiation - - - - 26 

15. Definition of Holomorphic Functions. Simply-connected 

Regions. Inverse Functions. Harmonic Functions - - 27 



x CONTENTS 

SECT. PAGK 

16. The Exponential Function - - - 32 

17. Circular Functions. Hyperbolic Functions 33 

18. The Logarithmic Function. The Inverse Tangent Function - 34 

19. The Transformation w = Log z - 35 

20. The Generalised Power 36 

21. Conformal Representation 37 

22. Singular Points. Poles. Meroinorphic Functions. Essential 

Singularities. Branch Points. Zeros. Theorems - 38 

Examples III. 40 



CHAPTER IV. 
INTEGRATION. 

23. Convergence and Uniform Convergence of Sequences ... 42 

24. Curvilinear Integrals. Differentiation under the Integral Sign - - 42 

25. Green's Theorem - 45 

26. Definite Integrals - 48 

27. Cauchy's Integral Theorem. Theorems on Integration 51 

28. Cauchy's Theorem : Alternative Proof - 54 

29. Cauchy's Residue Theorem. Residue at Infinity - - 57 

30. Evaluation of Definite Integrals. Theorems on Limiting Values 

of Integrals. Principal Value of an Integral - 59 

31. Cauchy's Integral 67 

32. Liouville'e Theorem - 68 

33. The Fundamental Theorem of Algebra 69 

34. Differentiation and Integration under the Integral Sign - 69 

35. Derivatives of a Holomorphic Function 70 

Examples IV. 71 

CHAPTER V. 

CONVERGENCE OF SERIES : TAYLOR'S AND LAURENT'S 

SERIES. 

36. Convergence of Series. Absolute Convergence. Multiplication 

of Series. Ratio Tests. The Hypergeometric Series 76 

37. Convergence of a Double Series - - 78 

38. Power Series. Circle of Convergence. Abel's Test. Multipli- 

cation of Power Series 80 



CONTENTS xi 

SECT. PA GE 

39. Taylor's Series - 82 

40. Laurent's Series. Principal Part at a Pole 84 

41. Fourier Series. Periodic Functions 86 

42. Classification of Uniform Functions. Integral, Rational Integral, 

and Transcendental Integral Functions. Theorems 88 

Examples V. - 89 



CHAPTER VI. 
UNIFORMLY CONVERGENT SERIES : INFINITE PRODUCTS. 

43. Uniformly Convergent Series. Continuity. Integration. Dif- 

ferentiation. Weierstrass's M Test 92 

44. Uniform Convergence of Power Series. Undetermined Co- 

efficients 95 

45. Additional Contour Integrals 97 

46. Legendre Polynomials. Expression in Definite Integral Forms. 

Recurrence Formula 99 

47. Expansion of cot z in a Series of Fractions 103 

48. Mittag-Leffler's Theorem. Weierstrass's Zeta Function. Weier- 

strass's Elliptic Function - - - 105 

49. Infinite Products. Unconditional Convergence - 107 

50. Weierstrass's Theorem. The Gamma* Function. Weierstrass's 

Sigma Function ... 108 

Examples VI. 109 



CHAPTER VII. 
VARIOUS SUMMATIONS AND EXPANSIONS. 

51. Expansions in Series by means of Residues. Theorems on 

Limiting Values of Integrals - - - - - - - 113 

52. Summation of Series by means of Residues. Gauss's Sum - 116 

53. Theorems on Roots of Equations - - - - - - 118 

54. Lagrange's Expansion. Rodrigues' Formula. Integrals involv- 

ing Legendre Coefficients - - - - - - - 119 

55. Analytical Continuation. Theorems. Recurrence Formula for 

Legendre Coefficients - 122 

56. Abel's Theorem - - 125 

Examples VII. 128 



xii CONTENTS 

CHAPTER VIII. 
/-GAMMA FUNCTIONS. 

SECT. PAGE 

57. The Bernoulli Numbers. Expression as Definite Integrals 132 

58. Asymptotic Expansion of Euler's Constant .... 134 

59. Convergent Integrals 136 

60. Uniformly Convergent Integrals. Differentiation and Integration 

under the Integral Sign - - - - - - - . 137 

61. The Gamma Function. Gauss's Definition. Formulae. The 

Function ^(z). Gauss's Function 11(2). Euler's Definition. 
Expression as a Contour Integral. Gauss's Theorem - 141 

62. The Beta Function. Duplication Formula for the Gamma 

Function - 144 

63. Asymptotic Expansion of the Gamma Function. Analytical 

Continuation of the Hypergeometric Function 146 

Examples VIII. - 154 



CHAPTER IX. 

INTEGRALS OF MEROMORPHIC AND MULTIFORM FUNCTIONS : 
ELLIPTIC INTEGRALS. 

64. Integrals of Meromorphic Functions. The Logarithmic Function 160 

65. Integrals of Multiform Functions. The Inverse Sine Function - 161 

66. Legendre's First Normal Elliptic Integral. Inversion of the 

Integral 163 

67. The Weierstrassian Elliptic Integral. Inversion of the Integral 167 

68. Elliptic Integrals in General. Legendre's Normal Integrals 169 

69. Complete Elliptic Integrals. Landen's Transformation 173 

70. Legendre's Relation - 175 

Examples IX. 176 



CHAPTER X. 
WEIERSTRASSIAN ELLIPTIC FUNCTIONS. 

71. Doubly- Periodic Functions. Theorem on Primitive Periods. 

Congruent Points. Period-Parallelograms 179 

72. Elliptic Functions. Weierstrass's Elliptic Function. Theorems 

on Poles and Zeros of Elliptic Functions 180 



CONTENTS xiii 



73. Relation between #>(z) and p'(z). The Weierstrassian Elliptic 

Integral 183 

74. The Addition Theorem for ^(2) - 185 

75. Properties of Weierstrass's Zeta Function. Legendre's Relation. 

Expression of Elliptic Functions in Terms of Zeta Functions - 187 

76. Properties of the Sigma Function. Expression of Elliptic 

Functions in Terms of Sigma Functions - 189 

Examples X. . - 191 

CHAPTER XI. 
JACOBIAN ELLIPTIC FUNCTIONS. 

77. The Values of $>(w; oi lf o> 2 ) when Wj is Real and o> 2 is Purely 

Imaginary - - 194 

78. Geometric Application - 196 

79. The Jacobian Elliptic Functions sn u, en u, dn u. Periods. Poles 198 

80. The Addition Theorems for sn u, en u, dn u - 202 

81. Jacobi's Imaginary Transformation - - 205 

Examples XI. 206 

CHAPTER XII. 
LINEAR DIFFERENTIAL EQUATIONS. 

82. Continuation of a Function by Successive Elements - - - 208 

83. Homogeneous Linear Differential Equations of the Second Order. 

Existence of an Integral in the Domain of an Ordinary Point. 
Integrals at Infinity. Analytical Continuation of the Integral 209 

84. Solution by Infinite Series, Legendre's Equation. Legendre's 

Function of the First Kind 213 

85. Theorems on Fundamental Systems of Integrals - 215 

Examples XII. - - 217 

CHAPTER XIII. 
REGULAR INTEGRALS OF LINEAR DIFFERENTIAL EQUATIONS. 

86. Integrals in the Neighbourhood of a Singularity. The Funda- 

mental Equation. The Associated Fundamental System 219 

87. Regular Integrals. Conditions for Regular Integrals - 222 

88. The Method of Frobenius. The Indicia! Equation. Solutions 

Free from Logarithms ........ 225 



xiv CONTENTS 

SECT. PAOE 

89. The Gaussian pifferential Equation. The Differential Equation 

of the Quarter Periods of the Jacobian Elliptic Functions 228 
Examples XIII. - 232 

CHAPTER XIV. 

LEGENDRE'S AND BESSEL'S EQUATIONS: EQUATIONS OF 
FUCHSIAN TYPE. 

90. Legendre Functions of the First and Second Kinds. Recurrence 

Formulae .... . 234 

91. Bessel Functions of the First and Second Kinds. Recurrence 

Formulae. The Function GJz). Relations between the 
Bessel Functions. The Zeros of J n (z) - 236 

92. Equations of Fuchsian Type - 243 

93. Riemann's P-f unction. The Twenty-four Integrals of the 

Hypergeometric Equation. Analytical Continuation of the 
Hypergeometric Function .... . 244 

94. Spherical Harmonics. Legendre's Associated Functions of the 

First and Second Kinds - 249 

Examples XIV. 252 

CHAPTER XV. 

SOLUTION OF DIFFERENTIAL EQUATIONS BY DEFINITE 
INTEGRALS. 

95. First Method of Solution. The Branch Points of the Integral - 255 

96. Gauss's Equation. Definite Integral Form of the Hypergeo- 

metric Function - - 258 

97. Legendre's Associated Equation. Expressions for P n m (z) and 

Q n m (z) .--- 259 

98. Second Method of Solution 266 

99. Bessel's Equation. Definite Integral Expressions for J w () and 

G n ( Z ) --- 266 

100. Asymptotic Expansions of G n (z) and J M () 271 

Examples XV. 275 

MISCELLANEOUS EXAMPLES - " - 277 

INDEX - 294 



CHAPTEE I. 



COMPLEX NUMBERS. 

1. Definition of Complex Numbers. A number of the form 
p + iq, where p and q are reai.and i is a root of the equation 
i z -fl=0, is called a Complex Number. If g = the number 
is said to be purely real, and if > = it is said to be purely 
imaginary. The complex numbers p + iq and p iq are called 
Conjugate Numbers. The number p + iq is zero if and only if 
p = and q = 0. 

If p=p + iq } it is frequently found convenient to write B(p) for p and I(p) 
for q, where B()o) stands for the real part of p and I(/o) for the imaginary 
part of p. 

Complex Numbers are subject to the same algebraical laws 
of addition, subtraction, multiplication, and division, as real 
numbers. These operations, when applied to real and complex 
numbers, produce real and complex numbers only; and it will 
be shewn ( 6, 20) that this is also true of the remaining 
algebraical operation of root extraction. 
YA 



X' O 
Y' 




FIG. 1. 



2. Geometrical Representation of Complex Numbers. The 

Complex Number s = x + iy can be represented geometrically by 



2 FUNCTIONS OF A COMPLEX VARIABLE [CH. i 

means of a Rectangular Coordinate System X'OX, TOY (Fig. 1). 
The point P(x, y) corresponds uniquely to the number z, and is 
called the point z. In particular, points on the #-axis correspond 
to purely real numbers, and points on the ^/-axis to purely 
imaginary numbers. The figure is called the Argand Diagram, 
and the coordinate plane is spoken of as the 0-plane. 

Example. If z 1 and z 2 are conjugate numbers, shew that the straight line 
joining the points z 1 and 2 2 is bisected at right angles by the #-axis. 

3. Modulus and Amplitude. In polar coordinates P is the 
point (r, 0), where r denotes the positive value of OP, and 
the angle XOP. The angle XOP is defined as the angle 
traced out by a radius-vector which revolves either positively 
or negatively from its initial position along OX till it coincides 
with OP. OP or r is called the Modulus of 0, and is written 
mod or | z \ ; is called the Amplitude * of 0, and is written 
amp 2. The amplitude can evidently have an infinite number of 
values differing from each other by multiples of 2-Tr: that value 
which satisfies the inequalities 

-7r<0^7T 

is called the Principal Value of amp z. 

The rectangular and polar coordinates are connected by the 
relations x = rcosO, y = r sin 0, 

r = Jx 2 + y\ tan = y/x. 
From these it follows that 

z x + iy = r (cos 4- i sin 0), 

an equation which expresses z in terms of its modulus and 
amplitude. 

Example 1 . Prove | cos + i sin 6 \ = 1 . 
^ Example^. If z = x + iy, shew that |#|^|z| , \y |^| z\. 

Vectors. A line AB (Fig. 1), equal to, parallel to, and in the 
same direction as OP, may also be used to represent the number z ; 
mod (AB) and amp (AB) are then identical with z \ and amp z. 
AB is called a Vector. It follows that 

BA=-AB. 

* The word Argument is used by some writers in place of Amplitude. 



3, 4] MODULUS AND AMPLITUDE 3 

4. Geometrical Representation of Addition. Let P l and P 2 

(Fig. 2) be the points z l = x l -{-iy l and z z = x z -\-iy z . Then 

YA R, 




FIG. 2. 



Through Pj draw P 1 P 3 equal to, parallel to, and in the same 
direction as OP 2 . P 3 has coordinates (fl? 1 + a? 2 , yi + y 2 ), and is 
therefore the point z^-{-z 2 . In vectorial notation 

OP = 



3 = O?! + OP 2 = OP 2 

Subtraction. Since X 2 = z l + ( 2 ), a subtraction can always 
be treated as an addition. Thus, if P 3 (Fig. 2) is the point % 



THEOREM I. The modulus of the sum of any number of 
complex quantities is less than or equal to the sum of their 
moduli : that is, if n is any positive integer, 

|^ + 2 +...+0 n ^l^l + l^l + .-. + l^n . 

This follows from the geometrical theorem that a side of a 
triangle is less than or equal to the sum of the other two sides : 
thus (Fig. 2) 

mod (OP 8 ) ^ mod (OPj) + mod (P^g). 

Therefore K + ^l^l^ | + K| . 

Hence z+2 + 2f = 2 + z + z 



and so on. 

THEOREM II. The modulus of the difference of two complex 
quantities is greater than or equal to the difference of their 
moduli. 

The verification of this theorem is left as an exercise to the 
reader. 



FUNCTIONS OF A COMPLEX VARIABLE [CH. i 

5. Multiplication and Division. Let z 1 = r 1 (cos0 1 +isin0 1 ), 

z 2 = r 2 (cos 2 + i sin 2 ), . . . , z n = r n (cos 6 n + i sin 
Then, by De Moivre's theorem, 



, Hence, the modulus and amplitude of a product are equal 
respectively to the product of the moduli and the sum of Ihe 
amplitudes of the factors. 

In particular, if n is a positive integer, and if z = r (cos 6 + i sin 0), 

then z n = r w (cos nO + i sin 710). 

Example. If p + iq is a root of the equation 

a 2 w + a ] 2 M - 1 + ... + a n = 0, 
where the coefficients , a t , ... , a n are real, prove that p -iq is also a root. 

Again, ^ = ^ {cos (0 X - 2 ) + i sin (0 t - 2 )} ; 

%2 f% 

so that the modulus and amplitude of a quotient are respectively 
the quotient of the moduli and the difference of the amplitudes 
of the numerator and denominator. 
It follows that the equation 

z n = r n (cos nO + i sin n6) 
holds when n is a negative integer. In particular, 

mod (1/0) = I/ 10 1 and amp (1/0)= amp0. 
v Example 1. Give a geometrical construction for l/z. 
J Example 2. Shew that amp ( } = BAG. 

VAB/ 

Let OP and OQ be parallel to and in the same direction as AB and AC. 



Then amp (==} = amp AC - amp AB 

= ampOQ-ampOP 



= BAC. 



If the angle so obtained is a positive {Fig. 3 (a)} or a negative (Fig. 3(6)} 
reflex angle, the principal value of the amplitude of the quotient is obtained 
in the first case by subtracting and in the second case by adding 2?r ; the 
resulting amplitude is in the first case negative and in the second case 
positive. As a rule, when the amplitude is mentioned, it is to be understood 
that the principal value is referred to. 



5, 6] 



MULTIPLICATION 



. / \ / \ 

*/ Example 3. Shew that, if amp ( ^ ? ) = amp ( -? ) , the points 2 3 and 2 4 

\z 1 z 3 / \Zi zJ 

are on the same side of the line joining z l and z 2 , and 1? 2 2 , ^ 3 , ^ 4 , are coney clic. 
Let Pj, P 2 , P 3 , and P 4 be the points z lt z 2 , z 3 , and 2 4 respectively. Then 



amp 

Therefore P 1 P 3 P 2 = P 1 P 4 P 2 . 

J^Ioreover, the points P 3 and P 4 must be on the same side of the line 
for if not, the angles P!?^ and P^^P^ would have opposite signs. 
Hence the points P 1} P 2 , P 3 , and P 4 are coney clic. 




Fio. 3. 



6. Root Extraction. If n is a positive integer there are n 

distinct values of z n . 

For, since, if K is any integer, 



/ O + ZKTT . . + 2/c7rY* . . 

1 cos \-^ sin - ) = cos + ^ sin 6, 
\ n n / 



it follows that r n (cos 



+ 2/C7T 



-fisin 



2*-7T\ 

) is 
I / 



an 7i th root of 



n n 

Now, if for AC the numbers 0, 1, 2, 3,..., 

i 

Ti1, are substituted in succession, n distinct values of z n are 
obtained. The substitution of other integers for K merely gives 
rise to repetitions of these values; and there can be no other 

values, since z n is a root of the equation x n = z, which has not 
more than n roots. 

Similarly, if p and q are integers, and q is positive, 



where K = 0, 1, 2, . . . , q 1. 



6 FUNCTIONS OF A COMPLEX VARIABLE [OH. i 

/ Example. Shew that the n th roots of any number can be represented by 
n equidistant points on a circle with centre at the origin. 



EXAMPLES I. 

1. Shew that the straight line joining the points z l and z 2 is divided in 
the ratio m : n at the point (mz 2 + nz-^l(m + ri). 

2. Prove that the centroid of the triangle whose vertices are z lt z 2 , and 

z 3 is(z 1 + z 2 +z 3 )/3. 

N/ 3. Prove that the modulus of the quotient of two conjugate numbers 
is unity. 

4. Prove that amp z - amp ( - z) = TT according as amp z is positive or 
negative. 

v/5. If | z 1 1 = j z 2 1 , and amp2 1 + amp^ 2 = 0, shew that z l and z 2 are conjugate 
numbers. v 

* 6. If 2 cos = a + lfa, shew that 2 cos n6=a n + l/a n . ~~^~* /U 

7. Prove algebraically that | z 1 +z 2 \ ^> \ z l \ + | z 2 \ . 

8. Shew that, if \z 1 + z 2 + ...+z n \ = \z 1 \+ z 2 \ + ... + \z n \, the Z'B must all 
have the same amplitude. 

9. Shew that, if amp | p ~ z v( z i ~ Z V j- = TT, then z 3 and z 4 are on opposite 
sides of the straight line joining % and z 2 , and z lt z 2 , z 3 , 2 4 , are coney clic. 

10. Let A, B, C, and D be the points z lt z 2 , z 3 , and 4 . Shew that, if 
z^z 2 + 2 3 2 4 = and z 1 + z 2 = 0, then A, B, C, and D are coney clic and the triangles 
AOC and DOA are similar. 

11. If AC : CB : : - AD : DB, and if A, B, C, D are the points z l5 z 2 , z 3 , z 4 , 
shew that A, B, C, an d D are coney clic, and prove (z l + z 2 ) (z 3 + z 4 ) = 2 (z^z 2 + z 3 2 4 ) : 
also prove triangles AOC and DOA similar, where O is the mid-point of AB. 

12. Prove that the two triangles whose vertices are the points a l , a 2) a 3y 
and b lt 6 2 , 5 3 , respectively, are directly similar if and only if 

= 0. 



13. Prove that the curves - constant and amp( -j = constant 
are orthogonal circles. 

^/ 14. Prove that the imaginary n ih roots of a real quantity can be arranged 
in conjugate pairs. 
\/ 15. Picture on a diagram the roots of the equation s 5 + 1 = 0. 

16. Shew that the equation 3 < 2,z = (z + l) 6 has four complex roots, two of 
which lie in the second quadrant and two in the third. Shew that all the 
roots lie on a circle. 

(See also Miscellaneous Examples, 1-9.) 



CH. ii, 7, 8] 



CHAPTEE II. 
FUNCTIONS OF A COMPLEX VAEIABLE. 

7. Uniform Functions. When a variable complex quantity 
w is connected with another variable complex quantity z in such 
a way that to each value of z there corresponds one value of w, 
w is said to be a Uniform or Single-valued function of z. For 
example, a polynomial in 0, or the ratio of two polynomials, is 
a uniform function of z. The formal definition of a Holomorphic 
function of a complex variable will be given in Chapter III. 

The values of 0, for which w is a function of z, may be limited 
to some assigned region of the plane. Thus the equation 



where x is real, defines y as a function of x for those values of x 
and those alone which satisfy the inequality 1 < x < 1. 

Multiple-valued Functions. If several values of w correspond 
to each value of z, w is said to be a Multiple-valued or Multiform 
function of z. For example, >Jz is a two-valued, and V0 an 
^-valued function of z. 

Path of Variation. In the theory of functions of a real 
variable, the independent variable x can only vary by values which 
correspond to points on the #-axis : in the theory of functions of 
a complex variable, on the other hand, the independent variable 
z can vary by values corresponding to the points of any path 
connecting the initial and final points. 

8. Transformations. If w is a function f(z) of z, the relation 
between w and z may be interpreted geometrically, and the 
relation may then be called a transformation : the point z is said 
to be transformed into the corresponding point or points w by 
means of the transformation w=f(z). If iu = az + b, the trans- 
formation is called a linear transformation. If W<f>(z)/\fs(z), 



8 



FUNCTIONS OF A COMPLEX VARIABLE 



[CH. II 



where 0(0) and \fr(z) are polynomials, the transformation is said 
to be rational. Transformations of the type w = (az + b)/(cz + d) 
are known as bilinear transformations. 

We proceed to investigate the geometrical meaning of linear 
and bilinear transformations. 

I. w = z + b. Let P, Q, and B (Fig. 4) be the points z, w, and 
b. Then, since PQ = OB, it follows that the effect of the trans- 
formation is to impose on every point z a translation equivalent 
in magnitude and direction to OB. 

YJ 




'P i 



FIG. 4. 



II. w = az. This transformation gives | w \ = \ a \ . \ z , and 

amp w = amp a -f- amp z. 

Consequently, if P and Q are the points z and w, the point 
Q can be derived from the point P by turning the radius-vector 
OP through an angle amp a and then multiplying it by | a \ . It 
follows that any figure in the plane is changed by the trans- 
formation into a similar figure. 

III. w = az-\-b. This, the general linear transformation, can 
be effected by applying transformations II. and I. in succession. 
X<ike transformation II. it transforms any figure in the plane 
into a similar figure. The ratio of the distances of corresponding 
points is given by the equation 



and the angle between corresponding lines by the equation 

amp(to 1 w 2 ) amp(0 1 z z ) = amp a. 
IV. w=l/z. Here |w]=l/|0f, and amp w = amp 0. Now 



8] 



TRANSFORMATIONS 



let P (Fig. 5) be the point z and P' the inverse of P with regard 
to the circle [0| = 1. Then the modulus of P' is l/\z\ and its 
amplitude amp z. Again, let Q be the image of P' in the cc-axis ; 
then the modulus of Q is 1/|0|, and its amplitude is amp z. 
Hence Q is the point w. It follows that the transformation is 
equivalent to an inversion in the circle of unit radius with the 
origin as centre, followed by a reflection in the #-axis. 




Point at Infinity. As z tends to infinity, w approaches the 
origin. In the theory of the complex variable, infinity is regarded 
as a point ; namely, that point which is related to the origin by 
means of the transformation w=l/z. 

V. w = a/z. This can be regarded as a combination of trans- 
formations IV. and II. 

VI. The general bilinear transformation w = (az+b)/(cz+d), 
where a/b =f= c/d. (If a/b = c/d, then w is a constant.) 

This transformation can be written 

V 

(bc ad)/c 2 a 



It can therefore be effected by combining the three 
transformations z 1 = z + d/c, z 2 = k/z lt where Jc = (bc ad)/c z , and 
w = 2 4- a/c. It should be noted that z can also be derived from 
w by the bilinear transformation z = ( dw + b)/(cw a). 

Since the inverse of a circle is a circle or a straight line, 

O ' 

it follows that bilinear transformations transform circles into 
circles or straight lines. 



10 FUNCTIONS OF A COMPLEX VARIABLE [CH. n 

V Example 1. Apply the transformation w = (22 + 3)/(2-4) to the circle 



Since w = 2 + ll/(z-4), the transformation can be effected by applying 
successively the transformations 

(i) ^ = 2-4, (ii) 22 = 1/2!, (iii) %=H%, and (iv) w= 
From transformation (i) we get 

#=#i + 4, y=^. 

Hence fo + 4) 2 + yi * - 4^ = 0. 

Transformation (ii) gives 



.Therefore 16(^ 2 2 +3/2 2 ) + 8^2 + 4y 2 + 1 =0. 

Again, from transformation (iii), 



so that 16 (> 3 2 + y 3 2 ) + 88^7 3 + 44y 3 +121=0. 

Finally, if w=u-}-iv, transformation (iv) gives 

#3 = ^-2, y 3 =-y. 

The given circle is therefore transformed into the circle 
I6u 2 + 16v 2 + 24w + 44v + 9 = 0. 

Example 2. Shew that the transformation of Example 1 changes the 
circle ^2+y 2 -437=0 into the line 4^ + 3 = 0, and explain why the curve 
obtained is not a circle. 

9. Geometrical Representation of Functions. It is often 
convenient to represent the dependent variable w on a different 
plane from the independent variable z. This plane is called the 
u>-plane, and w = u+iv is represented on it by the point (u, v) 
referred to rectangular axes U'OU, V'OV. If w is a uniform 
function f(z) of z> and if z moves from a to b by different paths in 
the 0-plane, w will move from /(a) to /(&) by different paths 
in the w-plane. In the case of multiple-valued functions, how-- 
ever, it will be shewn that the final point attained in the tu-plane 
depends on which value of w is selected as initial value, and 
also on the path followed by z in the 0-plane. 

Example 1 . Let w = z\ so that u x 9 -- y 2 , v = 1xy . 

Then, if #=0, u= -y 2 and v = 0. Hence as z moves up the y-axis from 
- co to 0, u increases from oo to 0, and therefore w moves along the w-axis 
from - oo to 0. Again, as z moves up the y -axis from to +00, u decreases 
from to oo , and therefore ^ moves back along the -M-axis from to oo . 

Similarly, it can be shewn that as z moves along the #-axis from - oo to 
+ oo , w passes along the w-axis from +00 to 0, and then back from to +00. 
Likewise, the positive and negative parts of the v-axis correspond respec- 
tively to the lines y x and y= x. 



9] 



GEOMETRICAL REPRESENTATION 



11 



Again, if we put z=r(cos 6 + isin 6} and w = p(cos<f> + isin <), we have 
p = r 2 and < = 20. 

Hence, if z lies on the circle ABCD (Fig. 6) of radius a, w will lie on the 
circle PQRS of radius a 2 . Let 0=0, < = initially, so that A and P are the 
initial positions of z and w. Then as z passes round the quadrant AB in the 
anti-clockwise direction, Q and <f> increase to 7r/2 and TT respectively, so that 

plane 

A LU - plane 




FIG. 6. 

w passes round the semi-circle PQR. Similarly, it can be shewn that, as z 
passes round the quadrants BC, CD and DA, w passes round the semi-circles 
RSP, PQR and RSP respectively. Thus, when z describes the circle ABCD 
once, w describes PQRS twice. 

\/ Example 2. If w = z 2 , and if z describes the line # = c, shew that w 
describes the parabola u = c 2 -v*/4c 2 . Trace on a figure, for the particular 
case c = 1. the course of w as z moves up the line x = 1 from - oo . to +00. 

In applications it is often important to trace the change in 
the amplitude of w when z describes a closed curve. We shall 
consider some particular cases. 

Y 



M 




( 1 ) w = z. Here amp w = amp z. Let z describe a closed curve 
LMN (Fig. 7) about the origin. Then, if z passes round LMN 



12 



FUNCTIONS OF A COMPLEX VARIABLE [CH. n 



once in the positive direction, amps, and consequently ampw, 
will increase by 27r. Similarly, if z passes round the curve once 
in the negative direction, amp z and amp w will each decrease by 
2?r; while n successive revolutions in the positive or negative 
direction will alter the amplitudes by + Znir or 2mr. 

Again, if the origin is exterior to the closed curve APBQ 
(Fig. 8) described by 0, the amplitudes of z and w will increase 
Y 




from ^.XOA at A to ^XOB at B, and then decrease from 1.XOB 
to ^XOA; so that the total change is zero. 

(2) w = a(z z l ), where a and z l are constants. Here 



so that, since amp a is constant, the change in amp w is equal to 
the change in amp(0 zj. Hence, if z describes a closed curve 
surrounding z l in the positive or negative direction, amp w will 
alter by +27r or 2?r; while, if z 1 is exterior to the curve, 
amp w will return to its original value. In the first case w will 
describe a closed curve in the w-plane about the origin ; while 
in the second case it will describe a closed curve not enclosing 
the origin. 

(3) w = a(z %)( Zz)(z 3), where a, z v z 2 , and z 3 are 
-constants. 

Here amp w = amp a + amp (z X ) + amp (z z z ) + amp (z z 3 ). 

If z passes round the curve C (Fig. 9), which does not contain 
any of the points z lt z z , Z B , then ampw will return to its 
initial value ; so that w will describe a closed curve not enclosing 
the origin. If z passes round C^, C 2 , or C 3 , amp w will be altered 
by 27r, 47r, or 6?r, and w will pass round the origin once, twice, 
or thrice as the case may be. 

(4) w = a(z z^)(z z z )...(z z n ). If in this case z describes a 
closed curve within which none of the points 15 2 , ..., z n lies, 



9] 



VARIATION OF THE AMPLITUDE 



13 



it follows, as in cases (2) and (3), that ampw will regain its 
initial value, and w will describe a closed curve which does not 
surround the origin ; while, if z describes a closed curve within 
which r of these points lie, amp w will be altered by 2r7r, and w 
will pass round the origin r times. 

(5) w = a(z-z 1 )/(z-z z ). 

Here amp w = amp a + amp (z z^) amp (z %). 
It follows that, if z describes the curve C x (Fig. 10) or C 2 in the 
positive direction, amp w is increased or decreased by 2?r ; while, 



O 





Fio. 



FIG. 10. 



if z describes either of the curves C or C 8 , amp0 regains its 
initial value. 

In all these cases it is obvious that the change in amp w due 
to the description of any closed curve is independent of the 
shape of the curve, so long as the same set of points z lt z 2 , z s , ... 
lies inside or outside it. It is often found convenient to take the 
curve in the form of a circle. 

(6) w = s/z. If z = r(cos + i sin 0), then w has two values, 

w^r^cos (0/2) + i sin (0/2)} 
and w 2 = r 1 / 2 {cos(0/2 + 7r) + 'i sin(0/2 + ?!)} = -w^ 

Each of these two quantities w l and w 2 varies with z, and is 
therefore a function of z : they are called the Branches of the 
two- valued function w. 

Let z start from the point P(r, a) (Fig. 11), and let the initial 
values of w 1 and w z be 

w = r 1 / 2 (cos (a/2) + i sin (a/2)} and w z = w r 
Then w^ and w 2 will be represented by the points P^r 1 / 2 , a/2) 
and P 2 (r 1 / 2 , a/2 -f TT) in the w-plane. Now, if z moves round the 
circle PQR of centre and radius r, 6 will increase by 2?r, and 



14 



FUNCTIONS OF A COMPLEX VARIABLE [OH. n 



amp w by TT. Consequently w l will move round the semi-circle 



PjQiRjP.2 and w z round the semi-circle P^R^ in the w-plane : 
the final values of w l and w 2 will be w z and w l . A revolution 
of 2 about the origin therefore interchanges the branches of w. 
Two such revolutions bring back w l and w z to their original 
values ; or, graphically expressed, if z moves round the circle 
PQR twice, w l and w z each move round the circle P 1 Q 1 P 2 Q 2 once. 



w- plane 



Q 




FIG. 11. 

If the circuit described by z does not enclose the origin, will 
regain its initial value a, and w l and w z their initial values w^ 
and w 2 . 

The point O is called a Branch Point of w, because a circuit 
about it interchanges the branches of the function. 

(7) w = -Ja(z z l ). Here amp w = \ amp a -f J amp (z X ). 

This is again a two-valued function. A single circuit about z 1 
interchanges the branches, while a double circuit brings them 
back to their initial values. On the other hand, the description 
of a circuit which does not enclose z l effects no alteration in the 
branches. Hence z l is a Branch Point of w. . 

Here amp w = J amp a + \ amp(0 z ) + i amp (2 z 2 ). 

Hence the description of C x (Fig. 12) or C 2 interchanges the 
branches, while the description of C or C 3 leaves them unaltered. 
Thus 0J and 2 are Branch Points of w. 

(9) w y(z a). If z a = r (cos -f- i sin 0), w has n branches 

!/ 0+2S7T 9 + 2STT\ 

w,, w 9) .... w n , where w s r n (cos h^sm -I. A 

\ n n / 

positive circuit round the branch-point a increases by 2?r, and 



9] 



BRANCH POINTS 



therefore changes u^ into w z , w 2 into w s , 

which do not enclose a leave the branches unaltered. 



15 

Circuits 




FIG. 12. 



Example 3. Let w = \/(l -z)(l +2 2 ), and let the value of w when z is at O 
be + 1. Then if z describes the curve OPA_(Fig. 13), where A is the point 2, 
shew that the value of w at A will be ~is/6. 

The three zeros of w are 1, i, and i. Let B, C, and D be the corre- 
sponding points, and through C and D draw CL and DM parallel to OX. 

Y 
C 




A X 



M 

FIG. 13. 



Let the moduli and amplitudes of BP, CP, and DP be r lt r 2 , r 3 , and < 1} </> 2 , 
< 3 , respectively, where z_XBP = ^> 1 , ^LCP = ^> 2 , and ^MDP = < 3 . Then 

/ 1 \T "2^. ^,, 



It has still to be determined which of the two possible values +TT or -TT 
is to be assigned to amp ( 1). Now, when 2 is at O, <f> l = 7r, </> 2 = ?r/2, (/> 3 = 7r/2 ; 
so that < 1 + < 2 + < 3 = 7r. Hence, if amp(-l) = 7r, amp^=?r at O; while, if 
amp(-l)=-7r, ampw = at O; but 10= +1 when z is at O, so that the 
latter value must be chosen. Therefore 



Now, as 2 passes from O to A, ^ decreases from TT to 0, </> 2 increases from 
-7T/2 to -tan" 1 -!, and <j> 3 decreases from ?r/2 to tan" 1 .}. Therefore at A 
amp ?0= -7T/2 ; also ^ = 1, r. 2 =\/5, r 3 = \/5. Hence 

w = x/5 { cos ( - 7T/2) + 1 sin ( - ~/2) } - - i \/5. 



16 FUNCTIONS OF A COMPLEX VARIABLE [OH. n 

10. Roots of Equations. In works on the theory of equa- 
tions it is shewn how, by means of Sturm's Theorem, it is possible 
to find the number of real roots lying between any two real 
values of the variable. We shall now shew how to find the 
number of real or complex roots of an equation which are 
contained in various regions of the 2-plarie. 

Consider the equation 



We assume that every equation has a root : a proof of this 
important theorem will be given later (33). It follows that 
f(z) can be put in the form 

^0 (Z-Zl)(z -%>... (2-Zn). 

If z be taken positively round a closed circuit in the 2-plane 
which encloses r of the points z lt z 2 , ... , z n , the amplitude of f(z) 
will be increased by 2r?r. Consequently the number of roots of 
f(z) = which lie inside a given circuit can be ascertained by 
det rmining the change in the amplitude of f(z) when z passes 
round the circuit. 

The following theorem will be found useful in locating the roots. 

THEOREM. If z be taken round any part of a large circle with 
the origin as centre and radius R, and if be the change in 




FIG. 14. 



amp z, the change in the amplitude of f(z) will differ from nO by 
a quantity which tends to zero as R tends to infinity. 

For f(z) = z n (a Q + ajz + a 2 /z* + . . . + a n /z n ). 

Hence amp/(0) = n amp z -\- amp (a + ajz + a 2 /z 2 + . . . + & n / 0n )- 

Now | ajz + a 2 /z* + . . . + a n /z 11 \ < p, 

where P = \ c^ j/R + | a 2 |/R 2 + ... +| On|/R n . 

Let R be chosen so large that /3<| I- Then the point 
...+a n /z n must lie inside a circle of centre a or A 



ioi 



ROOTS OF EQUATIONS 



17 



(Fig. 14) and radius p. If OP be a tangent to this circle, 
amp(a +a 1 /2+...+a 7l /z n ) differs from ampa by an angle rj, 
which is not greater than ^AOP, and which can be made as 
small as we please by increasing R, and thus decreasing p. 
That is, amp/(0) = n amp z + amp a Y\. 

Hence Lim amp f(z) = n amp z -\- amp a . 

Therefore, when R tends to infinity, the change in amp/(z) 
tends to n times the change in amp z. 

Example. Investigate the positions of the roots of the equation 



Let w = 4 + 2 3 + l, and let z describe a contour consisting of the three 
portions : 

(1) the .t'-axis from to + QO ; 

(2) the first quadrant of a circle of centre O and radius infinity ; 

(3) the y-axis from +00 to 0. 

(1) At points on the .v-axis, w = u + iv=at + x? + l, so that u = 
and v = Q. Hence, as z passes along the .r-axis from to + QO , 

the w-axis from 1 to +00, and therefore amp w remains constant and equal 
to zero. 

(2) On the great circle ampz increases by ?r/2, and therefore, by the 
theorem above, amp w increases by 2?r. 

(3) At points on the ?/-axis, u=y* + l and v= -y 3 . Hence w lies on the 
infinite curve LMN (Fig. 15), given by these equations, and as y decreases 



VI 





FIG. 15. 



FIG. 16. 



from +00 to 0, w passes along this curve from infinity below the u-axis to 
the point M(w = l) in the direction indicated by the arrows. Hence the 
initial and final values of amp?0 are equal, both being zero. 

The total change in amp w as z passes round the complete circuit is there- 
fore 2?r, and it follows that one and only one root of the equation lies in the 
first quadrant. 

Similarly it can be shewn that only one root lies in each of the other 
quadrants. 

Again, let z describe the contour OABCO (Fig. 16), where A and C are 
the points 1 and 2', and ABC is a quadrant of the circle || = 1. 



18 



FUNCTIONS OF A COMPLEX VARIABLE 



[CH. II 



Then, firstly, the description of OA gives rise to no change in amp u\ 
Next, for points on a circle of centre O and radius R, 

z = E (cos $ + i sin 0) = R(l - P + 2&)/(l + * 2 ), where t = tan ((9/2), 

' 



Accordingly, at points on ABC, z = (l + it)/(l it), so that 



Hence amp w = amp{(3- 12 2 + * 4 ) + z'(2 + 2 3 )}-amp(l -it)*. 

Now, as 6 varies from to ?r/2, t varies from to + 1 ; so that amp (1 - it) 
decreases by ir/4. Hence amp (1 it)* decreases by TT. 
Again, let = 3-12 2 + * 4 and ^ = 2^ + 2^. 
Then the curve given by these equations is of the form shewn in Fig. 17, 




FIG. 17. 

the arrows indicating the variation of the point (, rj) as t increases from 
oo to + oc . 

Now, when = 0, = 3 and 77 = 0, so that amp( + ir/) = ; also, when t = l y 
=8 and ?7 = 4, so that amp ( + 117) = 6, where is the angle in the 
second quadrant for which tan# = 1/2. Hence the change in arupw due 
to the description by z of the quadrant ABC is 

7r+(9=2 7 r-tan-- 1 (l/2). 

Finally, at points on OC, u=y* + l and y= -y 3 , so that w lies on the 
curve LMN (Fig. 15). When y = l t wia at the point K(?0 = 2 i), and 

amp w tan -1 (l/2) ; 

while, when y = 0, w is at the point M(w = l) and arapw = 0: so that the 
change of amplitude due to path CO is tan -1 (l/2). 

Hence the total change of amplitude due to the circuit is 2?r, and there- 
fore the root which lies in the first quadrant lies within the unit circle. 



10] 



ROOTS OF EQUATIONS 



19 



Similarly it can be shewn that the root in the fourth quadrant lies within 
the unit circle, while the other two roots lie outside it. 

Again, it can be shewn that all the four roots lie inside the circle |s| = 2. 

For, if z = 2(l+it)l(l-it), 



Now the curve 



25 - 102 2 + 9 4 , 77 = 76* - 44Z 3 , 



is of the form shewn in Fig. 18, the arrows indicating the variation of the 
point (, 77) as t increases from oo to + 00. But as amp z varies from - TT 





FIG. IS. 



to +TT, t varies from - oo to +00 , and therefore amp(f + ITJ) increases by 4?r 
Also amp{l/(l -it)*} increases by 4;r. Hence ampw increases by STT, and 
therefore all the four roots lie inside the circle. 



EXAMPLES II. 

1. If w and z are connected by the bilinear transformation 



and if the points u\ and ?/; 2 correspond respectively to the points z l and z 2 , 
shew that 



2. If iv = (az + b)/(cz + d), and if the locus of z is an arc of a circle 
standing on the chord joining the points z l and z 2 , shew that the locus of w 
is an arc of a circle standing on the chord joining ^ and w 2 . 



20 FUNCTIONS OF A COMPLEX VARIABLE [OH. 



3. If w = (az + b)l(cz + d), and if the points w lt w 2 , w 3 , and u\ correspond 
respectively to z lt z 2 , z 3 , and 2 4 , shew that 



w 1 - ?/; 3 w 2 - w 4 z 1 

4. Shew that the constants in the transformation w = (az + b)/(cz + d) can 
be so chosen that three arbitrary points w lt w^, and w 3 correspond respectively 
to three arbitrary points z lt z%, and z 3 . 

5. Find the bilinear transformation which makes the points a, 6, and c 
in the z-plane correspond respectively to the points 0, 1, QO in the w-plane. 

z a bc 

A ns. w = -- T - . 
zc b-a 

6. Find the bilinear transformation which makes the points 1, i, - 1 in 
the 2-plane correspond respectively to the points 0, 1, oo in the w-plane. 
Shew that the area of the circle z\=l is represented in the w-plane by the 
half- plane above the real axis. ATIS. w= i(z !)/(* + 1). 

7. Prove that the relation w = (l+iz)/(i+z) transforms the part of the 
real axis between z=l and z= 1 into a semi-circle connecting w=l and 
w 1. Also find all the figures which, by successive applications of the 
relation, can be obtained from the originally selected part of the #-axis. 

8. Let w=*J(2 2z + & 2 '), and let z describe a circle of centre z = l + i and 
radius \/2 in the positive direction. If z starts from O with the value \/2 of 
u\ what are the values of w 

(i) when z returns to O ; 
(ii) when z crosses the ^/-axis ? 

Ans. (i) - V2 ; (ii) V20 { cos (3:r/8 + A/2) + i sin (3;r/8 + A/2) } , where A is 
the angle in the second quadrant for which tan A= -3. 

9. Let w = N /(5 - 2z + 2 2 ), and let z describe a circle of centre 3 = 1 + 2*' and 
radius 2 in the positive direction. If z starts from the point + 1 with the 
value +2 of w, find the values of w at the first and second crossings of the 
^-axis. 

Ans. (i) \/2t/(20 + 8\/3){cos(7r/3 + a./2) + isin(Tr/3 + ./2)}, where a. is the 

angle in the second quadrant for which tan a.= 4 \/3 ; 
(ii) \/2^(20-8\/3){cos(27r/3 + ^/2) + isin (27r/3 + ^/2)}, where /3 is 
the angle in the second quadrant for which tan (3= -4 + >/3. 

10. If vP=z+l, shew that, when the point z describes the circle |l=c, 
each of the points w describes the Cassinian ?v*. 2 = c, where r t and r 2 are the 
distances of w from the points +1 and - 1. 

11. Shew that the equation 2^+2+ 1 = has one root in each quadrant, 
and that the root belonging to the first quadrant lies outside the circle ! z \ 1 
and inside the circle | z \ = 2. 

12. Shew that the root of z* + z + l =0 belonging to the first quadrant lies 
inside the square whose sides are ^ = 0, x\, y = 0, and y = l. 

13. Shew that the equation 2 4 + 4(l + i>+l = has one root in each 
quadrant. 



n] EXAMPLES II 21 

14. Shew that two of the roots of the equation z 5 -z + l6=0 have their 
real parts positive, and three their real parts negative. Also shew that all 
five roots lie outside the circle |^| = 1 and inside the circle |z| = 2. 

15. Shew that the only root of z+ 1021=0 inside the circle z\ = l is 
real and positive. 

16. Prove that z^ + lOz- 1 = has no root the modulus of which exceeds 2. 

17. If w = { (2 + i)z + (3 + 4i) }/*, shew that : 

(i) as (x, y) describes the circle x 2 +y^ = l positively, the point (u, v) 

describes the circle (w-2) 2 + (v-l) 2 = 25 negatively ; 
(ii) as (#, y) describes the circle sfi+y 2 4. Qy 12=0 positively, the 
point (u, v) describes the circle (i*-l/2) 2 -f (v-13/12) 2 =(25/12) 2 
negatively. 

18. Apply the transformation w = l/z, (i) to the set of straight lines 
through the point (a, 0), and (ii) to the set of circles with this point as 
centre : and shew that the set (i) is transformed into a set of coaxal circles 
through the points w=0, w = l/, while the set (ii) is transformed into a set 
of coaxal circles, of which these two points are the limiting points. 




[CH. Ill 



CHAPTER III. 
HOLOMOKPHIC FUNCTIONS. 

11. Limits. A single-valued function f(z) is said to tend to 
the limit L as z tends to the value z l if, corresponding to any 
assigned positive quantity e, however small, a positive quantity 
v\ can be found such that \f(z) L|<e for all values of z (except 
Zj) which satisfy the inequality | z z l | < 77. For brevity we write 

I 

L 



This condition can be represented geometrically as follows : if y (Fig. 19) 
be a circle in the w-plane of centre L and assigned radius e, a positive 



z - plane 



YI 



LU - plane 





X 

FIG. 19. 

quantity 77 can be found such that, so long as z remains inside the circle C 
in the 2-plane of centre z l and radius 77, the corresponding point f(z) in the 
w-plane will remain inside y. 

The limit L is clearly independent of the path by which z 
approaches z r 

The limit L has not necessarily the same value as /(^) : for, 
consistently with the definitions of 7, any arbitrary value can be 
assigned to the function at the point z r 

Limit at Infinity. If, corresponding to any positive quantity 
e, however small, a positive number N can be found such that 



11, 12] LIMITS AND CONTINUITY 23 

|/(0) L <e for |#|>N,/() is said to tend to the limit L as z 
tends to infinity : that is, Lim f(z) = L. 

Example. Lim 1 jz = 0. 

z >oo 

Infinite Limits. If, corresponding to any positive number N, 
however large, a positive number ^ cato be found such that 

\f(z) |> N for z l |< //, /(s) is said to/tend to the limit infinity 



as z tends to z r 

Example. Liml/0=co. 






The branches of multiple-valued functions generally tend to 
different limits as z tends to r 

If the limit L is a function L^) of z lt and if an rj can be 
found such that, for all points z^ in a given region, \f(z) L(0 1 )| <[ e 
provided | z z l \ < /, /(z) is said to tend uniformly to the limit 
L(0 1 ) in the region. 

12. Continuity. The function f(z) is said to be continuous 
at z 1 if /(^) has a definite value, and if Lim f(z) = f(z^). 



If f(z^) is infinite, f(z) has not a definite value, and is therefore 
discontinuous at the point z : . 

The condition for continuity can be expressed as follows: if, 

corresponding to any e* an 77* can be found such that 

/ 
-e for 



/(z) is continuous at z r 

A function is continuous in a region, if it is continuous at all 
points of the region. 

If f(z) has a definite limit at z l different from /(%), f(z) is said 
to have a Removable Discontinuity at z lt and the function can 
be made continuous by replacing the value at z l by the limit at 
that point. 

To investigate the continuity of a function at infinity, put 
= !/ and test for f=0. 

THEOREM 1. The sum of a finite number of continuous 
functions is a continuous function. 

* In this book e will usually be understood to represent an arbitrarily small 
positive quantity, and T? a positive quantity. 



24 FUNCTIONS OF A COMPLEX VARIABLE [CH. m 

THEOREM 2. The product of a finite number of continuous 
functions is a continuous function. 

THEOREM 3. The ratio of two continuous functions is 
continuous except for values of z which make the divisor 
zero. 

The verification of these three theorems is left to the reader. 
The proofs are almost identical with those for functions of a real 
variable. 

THEOREM 4. If f(z) is continuous and has the value I at z lf 
and if <j>(z) is continuous at I, <f>{f(z)} is continuous at z r 

For, \f(z) l <, if \ z z i\<^*i', an d e can be chosen so that 
> if |f-J|O. Now let g=f(z); then 



provided | z z l |< >/. Hence [f(z)} is continuous at z = z v 

THEOREM 5. The real and imaginary parts of continuous 
functions are continuous functions. 

For, if w = u + iv is continuous at z = z lt and if its value at 
that point is w 1 = u l + iv l , an r\ can be found such that, for 



Thus J{(u>-u 1 y t +(v-v l )*\<e', 

so that | u u^ | < e, | v v l | < e. 

Hence u and v are continuous functions at z z v 

13. Uniform Continuity. A function f(z) is said to be 
Uniformly Continuous in a given region, if, corresponding to 
any e, an 17 can be found such that, for every point z l in the 
region, \f(z) f(zj | < e, when \z %]<;; i.e. if /(0) tends 
uniformly to /(%) in the region. 

THEOREM. If f(z) is continuous in a given region, it is uni- 
formly continuous in that region. 

The proof of this theorem depends on the following Lemma : 

Lemma. If \f(z) -/K) |< e for \z-z l \<rj, then 

l/<X>-M)l<2e for |*-* 2 |<J* 
where 2 ^ 8 an y poi n t interior to the circle \z z l = | y. 



13] 



UNIFORM CONTINUITY 



25 



Let G! (Fig. 20) and C 2 be the circles \z - z l = >/ and 
Then, if z and z 2 lie within C v 

!/(*) -M) I = !/(*) -M) +M) -/W I 



But if z 9 be restricted to lie within C 2 , every point z such that 
^ J>; will lie within C r Hence the Lemma holds. 




Fir,. 20. 

Now suppose that f(z) is not uniformly continuous in a region 
within which it is continuous. Divide the region into smaller 
regions by means of sets of equidistant lines parallel to the two 
axes. In one, at least, of these smaller regions f(z) is not uni- 
formly continuous. Divide this smaller region into still smaller 
regions in the same way as before. In at least one of these 
regions f(z) is not uniformly continuous. By continuing this 
process a series of rectangular regions is obtained, each of which 
is contained in the preceding one, and is a region of non-uniform 
continuity. Now let z^ be a point interior to all the regions of 
this series; then, since f(z) is continuous at z l , an ^ can be found 
such that |/(0) /(%)(< e/2, provided \z z^\<r\. Hence, by 



the Lemma above, \f(z)f(z z ) <e for 

interior to the circle I z 



], where z. 2 is 



\ ^rj\ so that this circle is a region 
of uniform continuity of the function. But if the subdivision of 
the given region be continued till a region of the series obtained 
above is reached which lies entirely within this circle, this 
rectangle is a region of non-uniform continuity for/(z). Also, 
since the rectangle lies within the circle, it is a region of uniform 
continuity for f(z). Thus two mutually contradictory results 
are obtained. Hence f(z) must be uniformly continuous in the 
given region. 



26 FUNCTIONS OF A COMPLEX VARIABLE [OH. m 

Functions of Two Real Variables. A function u(x, y) of x 
and y is said to be uniformly continuous in a given region if, 
corresponding to any e, an tj can be found such that, for every 
point (x, y) in the region, 

\u(x + Ax, y + Ay)-u(x, y)\<e, 

provided Ax \ < 17, | Ay | < r\. A function u(x, y) which is con- 
tinuous in a given region is also uniformly continuous in the 
region. The proof of this theorem is left as an exercise, to the 
reader. 

Again, let the continuous function u(x, y) have continuous 
partial derivatives of the first order in the region. Then, if 

+ Ay) u(x, y), 
u(x, y + Ay)} 

{u(x, y + Ay)-u(x, y)} 



~\ r-\ 

u(x + O^x, y + Ay) + Ay O, y + 2 Ay), 



where 0<0 1 <1, 0<0 2 <1. 
Now and are con 
therefore, from the property of uniform convergence, 



Now and are continuous in the given region, and 



where a |< e, | /3 |< e, provided | Aa5 1< 17, | Ay |< jy. 
Hence Ait = 



ox 

where a and ft tend uniformly to zero with AOJ and Ay at all 
'points in the given region. 

14. Differentiation. The Derivative of any function f(z\ 
obtained by applying a finite number of the algebraical operations 
considered in 4, 5, and 6 to z in succession, is 



These limits are obtained by the same rules as when the 
independent variable is real. It is important to notice that the 
value of the derivative is independent of the amplitude of Az. 



14, 15] DIFFERENTIATION 27 

Example 1. Prove -^- =M2 li ~ 1 , (i) for a positive integer, and (ii) for n a 
negative integer. 

Example 2. Prove -j- = nz n ~ l for TI a positive fraction. 
Let n=p/q, where p and q are positive integers ; then, if 
z = r(cos + i sin 6), 



~ / + 2rt?7r . . i/ -r ^-"- \ i 
2=r(cos hesm 1, where & = 0, 1,2, ...,^-1. 

Now let i=z ll , where f represents the branch of z l/ * corresponding to one 
particular -value of k. 

Then, if the increment Az of z correspond to the increment Af of , 



Hence 

dz qf9-i q q 

where the same value of z l/<Jt is taken on both sides of the equation. 
Example 3. Prove -j- = nz n ~ l for n a negative fraction. 

15. Holomorphic Functions. Any function of x and y can 
be regarded, according to the definition of 7, as a function of z : 
for if z be given, the corresponding values of x and y are known, 
and therefore the corresponding values of the function can be 
found. For example, one value of x iy or of x 2 y 2 corresponds 
to every value of z. But these functions cannot be expressed in 
terms of z, and it is much more satisfactory to regard them as 
functions of the two independent variables x .and y. 

Let w = u + iv, where u and v are real functions of x and y. 
Then, if z' = x iy, x = (z-\-z f )j% and y = (z z')/2i; so that u and 
v can be regarded as functions of the independent variables z 
and z'. Hence, if u and v are continuous f unctions % of x and y 
with continuous partial derivatives, the condition that w should 
be independent of z is 

'du'dx , c. .'dv?>x .'dv 3 



or _ _ -_ -cto J__ 

+1 %7 ~ 



and this is equivalent to the two equations 

'du'dv ?>u cw 






28 FUNCTIONS OF A COMPLEX VARIABLE [CH. nr 

Thus, if u and v are continuous, and possess continuous partial 
derivatives which satisfy equations (A), w is a function of x and 
y in which x and y occur only in the combination x-\-iy z\ it 
may therefore be expected that the function w, like the functions 
considered in 14, will have a derivative which does not depend 
at all on the way in which Ax and Ay tend to zero, i.e. which 

does not depend upon -f-. It will be shewn in the following 

doc 

theorem that this is the case. 

THEOREM. If iu = u + iv, where u and v are uniform con- 
tinuous functions which possess continuous partial derivatives, 
the necessary and sufficient condition that w should possess a 
definite continuous derivative is that these partial derivatives 
should satisfy equations (A). 

Let the increments Alt', Au, Av , and Az, of w, u, v, and z, 
correspond to the increments Ax and Ay of x and y. Then 

Aw _ Au + iAv 
Az ~ Ax -\-iAy 



Ax + i Ay 



where a, /3, a', /3', tend uniformly to zero with Ax and Ay. Thus 



div r . Aw 

- T -=Lim-: = , 

dz A2 _ >0 Az -, , .ay 

J- | t/ 7 

eta; 
Hence the necessary and sufficient condition that -^-. should be 

7 3 

CtlJ 

independent of -r- is 

'du . *dv _ 1 /du 

which is equivalent to equations (A). 



15] HOLOMORPHIC FUNCTIONS 29 

div 'dw 1 'dw 
COROLLARY 1. -?-==*-- 

dz ?)x i 3y 

COROLLARY 2. Since the partial derivatives of u and v are 

continuous, -j- is also continuous. 

az 

Definition. If a function is uniform and continuous, and 
possesses a definite continuous derivative at any point, it is said 
to be Holomorphic * at the point. 

A function is said to be Holomorphic in a given region, if it is 
holomorphic at all points of the region. 

Equations (A), expressed in terms of polar coordinates, become 



, ,. 



and the derivative is then obtained as follows : 
dw c)w 'dw "dr 'dw W 



sin 

= cos r +ir 



or \ or or./ r 

/ A ' ^ W 

= (cos 6 i sm 6) ^ 

Oi 

Example 1. Shew that the function e x (cosy + i sin?/) is holomorphic, and 
find its derivative. Ans. e x (cos 



Example 2. Shew that logr + i'# is holomorphic unless r=0, and find its 
derivative. A ns. (cos 6 - i sin Q)/r. 

Note. From the definition of a derivative the rules for 
differentiating products and quotients follow as in the case of 
the real variable. 

THEOREM. If f(z) is holomorphic in a given region, then, for 
all points z l in the region, 



where A tends uniformly to zero as z tends to z r 

The proof of this theorem is left as an exercise to the reader. 

*The words Regular and Analytic are used by some writers instead of Holo- 
morphic. The sense in which we shall use the word Analytic will be explained 
in Chapter XII. 82. 



30 FUNCTIONS OF A COMPLEX VARIABLE [CH. nr 



COROLLARY. If f(z) and <f>(z) are holomorphic, and if f(zj = 0, 
and 4>(z l ) = (\ while ^'( 



Function of a Function. If w=f(g) and =<f>(z) are holo- 
morphic functions of and z respectively, w is a holomorphic 

P (. dw dw d 

tunction of z\ tor ^ = - T? -f-. 

dz dg dz 

Simply-Connected Regions. If any two points in a region can 
be connected by a curve which lies entirely within the region, 
the region is said to be Connected. A connected region which is 
such that any closed curve lying entirely within it can be con- 
tracted to a point without passing out of the region is said to be 
Simply -Connected. Connected regions which are not simply- 
connected are said to be Multiply -Connected. The region 
enclosed by the curve C x (Fig. 21) is simply-connected, while 
the region between the curves C x and C 2 is multiply- connected. 




FIG. 21. 



The branches of multiple-valued functions can be treated as 
uniform functions in simply-connected regions which do not 
enclose any branch-points. No path in such a region can enclose 
a branch-point ; so that, after describing a closed path, the 
function regains its initial value. For example, -each branch of 
w = \/0 is holomorphic in the simply-connected region obtained by 
making the negative real axis a barrier which z cannot pass. 
Such a barrier is called a Cross-cut. The derivative l/(2/s), of 
course, takes the value corresponding to the value of *Jz under 
consideration. 

Inverse Functions. If w=f(z) is a holomorphic function such 
that w = w l corresponds to z = z l3 'z can be regarded as a function 
of w with z l corresponding to w l : if this function is uniform 
and continuous in a region of the w-plane which encloses w lt 



15] INVERSE FUNCTIONS 31 

then, since -y I -, -, it is a holomorphic function of w at all 
dw I dz , 

points of the region except those for which -y = 0. This 
function is called the Inverse Function of f(z). 

Example 1. If w = 2 , there corresponds to any value z l of z one value 
w 1 of w: conversely, one of the branches of z = *Jw gives the value z l of z 

corresponding to w = iv l . Now the only value of w for which -^-=0 is w = 0. 



Hence, if u\ + 0, w can be enclosed in a region in which the branch is 
holoraorphic, and therefore z = <Jw is the inverse function of w=z 2 . 

Example 2. For what values of z do the functions w defined by the 
following equations cease to be holomorphic ? 

(1) 2 = e"(cos v + i sin v) ; 

(2) z = log p + i(f>, where w = p(cos( + isin </>). 

Jws. (1) = ; (2) None. 

Laplace's Equation. It will be proved later (35) that if 
w = u-\- iv is a holomorphic function, w, u, and v have continuous 
derivatives of the second and higher orders. The reader can 
easily verify that u and v both satisfy Laplace's Equation 



The solutions of this equation are called Harmonic Functions, 
and are of great importance in Mathematical Physics. 

It follows that, if a function u or v is given, a corresponding 
holomorphic function w will not exist unless the given function 
is harmonic. If, however, this condition is fulfilled, the function 
w can be found by means of equations (A) : for example, if u is 
a uniform continuous function which satisfies Laplace's Equation, 
, 'dv .'dv -. 



is a complete differential, and v can therefore be found. 



Example. Shew that u=x z 3^/ 2 + 3.i i2 3?/ 2 +l is a harmonic function, 
and find the corresponding holomorphic function. 

A us. z* + 32 2 + 1 + ^C, where C is a real constant. 

Conjugate, Functions. If u + iv is holomorphic, u and v are 
called Conjugate Functions. These functions possess two im- 
portant properties : firstly, they satisfy Laplace's Equation ; and 
secondly, the curves u = c l , v = c 2 , where c and c., are arbitrary 
constants, intersect at right angles, since the product of their 



32 FUNCTIONS OF A COMPLEX VARIABLE [OH. m 



,. , . . mi. p 

gradients ( ~) and ( - / ) is 1. The systems of 
\dxj?iy/ \dxjcyy 

curves obtained by varying the constants c x and c 2 are called 
Orthogonal Systems. 

Example. Picture on a diagram the orthogonal systems given by 



16. The Exponential Function. The function u+iv, where 

u+ iv = e x (cos y + i sin y) 

is holomorphic for all finite values of z, since u, v satisfy 
equations (A), 15. When y is zero, the function becomes the 
ordinary exponential function e x : it is therefore regarded as 
the extension of e x to the domain of the complex variable, and 
is denoted by fcxp (z). Obviously 



Again, since 

i sin y) x e x ' (cos y' + i sin y') 



exp (z) x exp (Y ) = exp (z + z). 

Hence exp (z) x exp ( z) = exp (0) = 1 ; 

so that exp ( z) = 1/exp (z). 

Thus, exp(z) satisfies the index laws. It is often found 
convenient to write e z for exp (3): in particular, e iy stands for 
cos y + i sin y. 

Derivative, -j- (expz) = {e x (cosy + i sin y)} = exp z, 

az ox 

*W 

dz 

Periodicity. Since cos y and sin y have the period 2?r, exp (z) 
has the period Ziir : i.e. 

e z+2kin = gz( cos 2/C7T + i sin 2/c?r) = e z , 
where k is any integer. 

Zeros and Infinities. Since e z e x , e z can only have zero 
and infinite values when e x is zero or infinite. But e x is only 
zero when x ~ oo , and only infinite when x = + oo . The 
Exponential Function is therefore finite and non-zero if x is 
finite. 

Example. Shew that every period of exp (z) must be an integral multiple 
of Ziir. 



io, 17] CIRCULAR FUNCTIONS 33 

17. The Circular Functions. Since e ix = cosx + ismx and 
g - = cos x i sin x, 

e ixt e -ix e ix_ e -ix 

cos x ~ , sin 0;= ^- . 

2 2t 

These functions can therefore be extended to the domain of 
the complex variable by means of the equations 



COS2 = 77 , Sin 5 = =-: , 

* 2i\> 

which define them as holomorphic functions. 

r riie following well-known formulae can be derived from these 

definitions : 

sm 2 2+cos 2 2 = l ; 

sin (2^ + 2 2 ) = sin z 1 cos 2 2 + cos z l sin 2 2 ; 
008(0! + z 2 ) = cos z l cos z 2 sin z l sin z 2 ; 

d sin z d cos z 

.] = cos z 5 j = sin z ; 

sin( z}= sin z\ cos( z) = cosz. 

Note. If /( z) = f(z) for all values of z for which f(z) is 
defined, /() is said to be an odd function of z : if /( z) =f(z\ 
f(z) is an even function of z. Thus, sin z and cos z are odd and 
even functions respectively. 

Zeros. If sin 2 = 0, 



where k is any integer ; therefore is = iz + Skirl. 

Hence the values of z which make sin z zero are 0, TT, 2?r, 
3-7r, ---- 

Similarly, since e- /z = e- l ' s +< 2 *+ 1 > ir *, the values of z which make 
cos z zero are given by z = (k + J)TT, where k = 0, 1, 2, ____ 

The other circular functions are defined by means of sin z and 
cos 2: e.g. tan = sin z/cos z. The inverse functions are written 
sin" 1 s, tan" 1 ^, etc. 

The Hyperbolic Functions. These functions are defined by the 
equations : 

e z + e~ z . . e z e~ z sinh z 

cosh z = 9 ; smh z ^ > tanh 2 = ^ ; etc. 



A . , , .. . , ., 

Example. Prove: -- -. = cosli2:; j =8inhz; cosh-i-sinli-r 
dz <(~- 



M.K. 



34 FUNCTIONS OF A COMPLEX VARIABLE [CH. in 

18. The Logarithmic Function. If y = e x , where x and y 
are real, the inverse function is x = log y : for complex values of 
the variables, the inverse of the Exponential Function is defined 

as follows : '^ 

j 
Let z = r*(co8 + i sin 9) = exp (w) = e"(cos v + i sin v}. 

Then e 1 * = r, so that i& = log r and v = + 2&7T, where & is any 
integer. Hence the inverse function is 



where may have an infinite number of values differing by 
multiples of 2-Tr. This function is denoted by Log z. 

If z passes round the origin once in the positive direction, 
increases by 2?r and Log z by 2i?r. The origin is therefore a 
branch-point of Log z. Each of the infinite number of branches 
of Log z is uniform and continuous in the simply-connected region 
formed by taking a cross-cut along the negative real axis; and 
therefore, since it satisfies equations (A') of 15, it is holo- 
morphic in that region. That branch for which TT < 5: + TT 
is denoted by log z ; for positive real values of z this branch is 
the ordinary Naperian logarithm. 

Zeros and Infinities. Since log r is infinite when r is zero or 
infinite, Log z has infinities at the origin and infinity. Log z is 
only zero when both log? 1 and are zero ; i.e. when 0=1. 

Derivative. ogg = e -** j- (logr + ifl) = J 

d * arV ?K 

^.WV 

Example. Shew that Logos') = log z + log / + 2fe'. 

1 g*> _ ~ ! '"' 

Function tan~ l z. If = tan w = - -.- -. , then 

^ e lw +e~ M 



so that w = s~. Log f , where f = Now Log f is uniform if a 

2i \%z 

cross-cut is taken in the f-plane* along the negative real axis. 
But the transformation =(l + iz)l(Iiz) is bilinear, so that 
one point in the -plane corresponds to each point in the s-plane, 

*The notation ^=^ + ir] is adopted ; , rj, f, then correspond to x, y, z. 



18, 19] 



THE LOGARITHMIC FUNCTION 



35 



and conversely. Accordingly, if a cross-cut be taken in the 
0-plane corresponding to the cross-cut in the -plane, the function 
Log f will be uniform in the s-plane. Now, since 



to the part of the -axis between and 1 corresponds the 
2/-axis from i to +ix> , while to the ^-axis from 1 to oo 
corresponds the 2/-axis from ice to i. Hence, if a cross-cut 
is taken along these parts of the y-axis, the function 



tan~ 1 = ^ 



is uniform throughout the 3-plane. That branch which has the 
value zero when = is the Principal Value, and is equal to 

-r- -- -; its real part lies between Tr/2 and -rr/2, while 



- -- r- 

1 ~~%'Z/ 

its imaginary part varies from oo to + x . For any other branch 

1 . l 



where in is an integer. 
It follows that 



d 






2, and that 
1 



19. The Transformation w = Log z. Since u = log r, to circles 
run^tt.int in the 0-plane correspond lines u = constant in the 



z - plane 








v> 


, 




















, 


k, 


k. 


L, 


^ 


U' 

















v' 









VTT 

v=a 



Flo. 22. 



//-llane: the circles C , Cj, C. 2 , ... , C. lf C_ 2 , ... (Fig. 22) of radii 
1 '. ' , ,..,6"^, e" a , ... correspond to the equi-distant lines 

L , Lj, L. 2 , ... , L_j, L_ 2 , ..., 
equations are ^6 = 0, 1, 2, ... , 1, 2, .... 



36 FUNCTIONS OF A COMPLEX VARIABLE [OH. m 

To the origin and infinity in the z-plane correspond u oo 
and u = -h oo in the w-plane. 

Again, since v = 6, to the rays = constant in the s-plane 
correspond the lines v = in the w-plane ; so that, if a cross-cut 
be taken in the 0-plane along the negative #-axis, the entire 
s-plane is represented by that part of the w-plane which lies 
between the lines v = TT and v = + TT. If now the cross-cut be 
removed, and 6 increase from -w to STT, the entire 0-plane corre- 
sponds to the strip of the w-plane which lies between the lines 
v = TT and v = STT. Similarly the entire w-plane can be divided 
into strips of breadth STT, on each of which the entire z- plane is 
represented. Points in these strips which correspond to the same 
point in the -plane lie on the same parallel to the r-axis, at 
distances 2?r from each other. To each point in the i^-plane, 
however, corresponds only one point in the 2-plane, since 
exp (w) is a uniform function of w. Each strip of the w-plane 
represents one of the branches of w, the boundary in each case 
being assigned to the strip below it. 

Example. Shew that, for all values of m, 



- 

provided K( > 0. 

20. The Generalised Power. Up to this point z n has only 
been defined for rational values of n ( 5, 6). We are now in a 
position to define it for all values of n, rational or irrational, real 
or complex. 

If w Log 0, then z = exp (w) ; hence 

z = exp (Log 0) = exp (log z + 2&7ri), 

where k is any integer. Accordingly, for all values of n, we 
^define z n by means of the equation 

z n = exp (n log z + ZnltTri). 

COROLLARY 1. If n is an integer, z n lias only one value, 
exp (n log z), (cf . 5). 

COROLLARY 2. If n is a fraction pjq (q positive), z n lias q 
values given by 

exp ( log z] e**, where k = 0, 1, 2, . . . , q - 1 . 

^\q 

The reader can easily verify that this agrees with the results 
of 6. 



20, 21] CONFORMAL REPRESENTATION 37 

COROLLARY 3. If n is irrational or imaginary, z n has an 
infinity of values. 

Example 1. Prove -i-=nz n - 1 for all values of n, where the same value 
of z n is taken on both sides of the equation. 

Example 2. Shew that, for all finite values of rx, 



; > 

where s tends to infinity in any direction whatever. 
We have (15, Cor., p. 30) 



so that, if = l/, Lim 2 log(l + OL/S) = ou 

2->OC 

Thus, since the exponential function is continuous ( 12, Th. 4), 

Lira ( 1 + - Y = Lim e* l <* < 1 + a ' 2 > = e. 

2 _>oo \ Z J z _>. x 

See also Examples III. 13, 14, 15. 

21. Conformal Representation. Let w be a holomorphic 
function of 0; then, if the points w t w lt W 2 (Fig. 23), in the 
w- plane correspond to the points z, z l} z 2 , in the z-plane, 

T . w,w dw r . 
Lim = ^- 

2l ^ 2 z^-z dz 



T . w 2 w T . 0. 2 z 
or Lim = Lim . 

w l w z l z 

Hence, if the two triangles of vertices w, w lt w 2 , and z, z lt 0. 2 , 
w - plane z -plane 





FIG. '23. 



are infinitesimally small, they are directly similar. Also, since, 
to the first order of infinitesimals, 

x dw , 
(w l -w)= dz (z r -z\ 

the first triangle can be obtained from the second by turning it 
through an angle amp (dw/dz) and magnifying it in the ratio 



38 FUNCTIONS OF A COMPLEX VARIABLE [CH. m 

\dw/dz\. It follows that two intersecting curves in the 0-plaiie 
are represented in the w-plane by curves which intersect at the 
same angle. 

Each plane is said to be represented Conformally on the other. 
Examples of Conformal Representation have been given in 

9 and 19. The representation breaks down if -^- is either 
zero or infinite. 

Example. Deduce, from the principle of Conformal Representation, the 
theorem that the curves u = constant, v = constant, intersect at right angles, 
where u and v are Conjugate Functions. 

22. Singular Points. A point at which a function ceases to 
be holomorphic is called a Singular or Critical Point, or a 
Singularity of the function. For example, 2 = is a singularity 
of 1/z. 

If a circle can be drawn with the singular point as centre, so 
as to enclose no other singularity of the function, the singularity 
is said to be Isolated. The function I/sin (1/z) has a non-isolated 
singularity at z = Q: for, since sin(l/0) is zero for z = l/(Jc7r), 
where k is any integer, it is impossible to surround the origin 
with a circle which does not contain an infinite number of these 
points. 

A point which can be made the centre of a circle enclosing no 
singularity is called an Ordinary Point. If the radius of the 
circle is equal to the distance of the point from the nearest 
singularity, the interior of the circle is called the Domain of the 
point. 

Poles. If Lim (z z^) n f(z) = C, where C is a non-zero constant 



n a positive integer, z 1 is said to be a Pole of f(z) of order 
n, and f(z) </>(z)/(z Zi) n , where <f>(z) is holomorphic at s r If 
n = l, z l is a Simple Pole of f(z). For example, l/z n has a 
pole of order n at z = 0. 

Example. The function I/sin (B ZI) has a simple pole at z l : for ( 15) 

Lim ! (z -,) 7 M tJ- s =H - 7 - it =1. 
^V "sm (*-*,)/ \cos(* -*!)/, 

The function f(z) will have a singularity at infinity if =0 is 
a singularity of /(1/f). For example, az 2 +bz + c has a pole of 



22] SINGULAR POINTS 39 

the second order at infinity. If infinity is an isolated singularity 
of f(z), f=0 will be an isolated singularity of /(1/f), and a circle 
If | = e can be drawn to enclose no singularity of /(1/f) except 
f=0. Hence a circle \z =l/e can be drawn which will have 
within it every singularity of f(z) except infinity. 

Meromorphic Functions. A function which is holomorphic 
throughout a region except at isolated poles is said to be Mero- 
morphic in that region. 

Essential Singularities. If no value of n can be found such that 
Lim( z l ) n f(z) = C, then z l is said to be an Essential Singularity 

of f(z). Poles are Non- Essential Singularities. 

Example. The function e 1 -'-' has an essential singularity at 2 = 0. 

Branch-Points. The branch-points of multiple- valued func- 
tions are Singular Points : for example, z = Q is a singularity 
of Log z. 

Zeros. If f(z) = (z z l ) n (^(z\ where n is a positive integer and 
</>(z) is holomorphic and non-zero at z l} then z is said to be a Zero 
of f(z) of order n ; a zero of order 1 is also called a Simple Zero. 
If z l is a zero of f(z) of order n t it is a pole of l//(z) of order n. 

THEOREM 1. A pole is an isolated singularity. 

If z l is a pole of f(z) of order n, the function (z z l ) n f(z) is 
holomorphic at z l ; consequently, if C is its value at that point, 
an r\ can be found such that \(z z l ) n f(z) C\<^e ) provided 
| z z^ I < r\. Hence f(z) must be finite at all points except z l in 
the circle z z l \ = >/, so that the singularity is isolated. 

COROLLARY 1. The zeros of f(z) must also be isolated, or the 
function ~L/f(z) would have non-isolated poles. 

COROLLARY 2. If infinity is a pole of f(z), a circle can be 
drawn which encloses all the singularities of f(z) except infinity. 

THEOREM 2. No region can contain an infinite number of 
isolated singularities. 

Let a given region contain only isolated singularities, and let 
it be divided up as in 13. If there is an infinite number of 
singularities in the region, one at least of the divisions must 
contain an infinite number of singularities, and by continuing the 
process of subdivision a point can be found such that, in every 



40 FUNCTIONS OF A COMPLEX VARIABLE [CH. m 

neighbourhood of it, there is an infinite number of singular 
points, i.e. it is a non-isolated singularity, which is contrary to 
hypothesis. 

COROLLARY. If a function is meromorphic throughout the 
plane, and has an ordinary point or a pole at infinity, it follows 
(Th. 1, Cor. 2) that it has only a finite number of singularities. 

EXAMPLES III. 

1. Shew that l/{(z a)(z b)(z c)} is holomorphic except at , &, and c. 

2. Shew that the following functions are holomorphic, and find their 
derivatives : 

(i) e~ y (cos#+isin#). Ans. ie~ y (cos x + i sin x). 

(ii) cosh # cosy + ^sinh#siny. Ans. sinh x cos y + i cosh x sin y. 

(iii) sin # cosh y + i' cos #sinhy. Ans. cos # cosh y z'sin^sinli?/. 

(iv) cos x cosh y i sin x sinh y. Ans. sin x cosh y i cos x sinh y. 

3. If n is real, shew that r n (cosnd + isinn@) is holomorphic except 
possibly when r=0, and that its derivative is nr- 1 (cos n - 1 d + i sin n - 1 0). 

4. For what values of z do the functions w defined by the following 
equations cease to be holomorphic 1 

(i) z=e~ v (cosu + isinu). Ans. 2 = 0. 

(ii) z sinh u cos v + i cosh u sin r. Ans. z i. 

(iii) 2 = sin % cosh v + i cos w sinh v. Ans. z= 1. 

5. If <f> and T/T are functions of x and y satisfying Laplace's Equation, 

shew that s+it is holomorphic, where * a ? B *5~ o^ an( i ^ ==: ^T^"^^' 

6. Shew that w = e* (# cosy -y sin y) is a harmonic function, and find the 
corresponding holomorphic function. Ans. ze* + iC. 

7. If w=z 2 , shew that the curves u = c lt v = c 2 , are rectangular hyper- 
bolas, and represent them on a diagram for different values of c and c 2 . 

8. If z = sin u cosh v + i cos u sinh v, picture on a diagram the orthogonal 
systems uc^ v = c 2 . Shew that the first system consists of confocal 
hyperbolas, and the second of confocal ellipses. 

9. Shew that, (i) sin iz = i sinh z, (ii) cos iz = cosh z. 

10. Prove (i) sin (z 1 iz z ) = siuz l cosh z 2 i cos z^^ sinh z. 2 , 

(ii) cos (z l iz 2 ) = cos z\ cosh z. 2 T i sin 2 X sinh z. 2 . 

11. Prove (i) | sin (x ty) | = *J (sin 2 # + sinh 2 y} = cosh y 



(ii) | cos (# + ^y) | = x /(cos 2 x -\- sinh 2 y} cosh y 

Sij sinhy |. 

12. Prove Log ( - 1 ) = (2^- + 1 ) TT/, where -1 is any integer. 

13. If w is real and z = re ie , shew that 2 n = r" e ni ^ +2{ ' n \ where r n is real 
and positive. 



22] EXAMPLES III 41 

14. Shew that z n z n = z" +n ' for all values of n and ri, where suitable 
branches of the functions are taken. 

15. Shew that (z n ) n ' =z nn> for all values of n and ri, where suitable branches 
of the functions are taken. 

16. If w={(z-c)/(z + c)} 2 , where c is real and positive, find the 'areas of 
the 2-plane of which the upper half of the w-plane is the conformal 
representation. 

Ans. (i) The lower half of the circle \z =c ; (ii) that part of the plane 
above the #-axis which is exterior to the circle | z \ = c. 

17. If 10= -iccot(zf2\ shew that the infinite rectangle bounded by #=0, 
# = TT, #=0, # = 00, on the 2-plane is conformally represented on a quarter of 
the w-plane. 

18. Shew that infinity is a simple zero of (az 2 + bz + c)/(lz 3 + mz 2 +nz+p). 

19. Shew that the ratio of two polynomials is a meromorphic function. 

20. Shew that sees, cosec?, taii, and cot?; are meromorphic in the 
finite part of the plane. 

21. If w = sin -1 2, shew that w = /nr : Fi Log {1^ + ^(1 -z 2 )} according as the 
integer k is even or odd, a cross-cut being taken along the real axis from 
1 to oo and from - x to -1 to ensure that Log{?^-f x /(l -z 2 )\ should be 
uniform. 

Deduce that 7 - sin" 1 z = --j- - ^ , 

dz V(l-s-) 

where the branch of -j- - is chosen which corresponds with the branch 
of sin- 1 .* under consideration. 



22. Prove (i) Ljm-* i; (ii) Lta = - 2 ; 

z-n (-1)" , 
(in) Lim -- = , where n is an integer ; 

' 



x T . tan A: . 
(iv) Lim -= A. 

c-0 - 

23. Shew that all the values of i l are given by g-O^+iJ*, where / is any 
integer. 

24. If w=\jz, shew that the curves it = c lt v = c t , are orthogonal circles 
which pass through the origin, and have their centres on the ?/-axis and 
^r-axis respectively. 



[CH. IV 



CHAPTER IV. 
INTEGRATION. 

23. Limit of a Sequence. Let z lf z 2 , z s , ... be an infinite 
sequence of real or complex numbers; the sequence is said to 
converge to a limit I if, corresponding to any assigned e, a 
number m can be found such that z n < e, when n ^ m. 

If z n x n + iy n a,udl = a + ib, then \x n a < e and y n b <e; 
hence it follows that the sequences x lt x 2 , ~x s , ... and y lt y%, y 3 , ... 
converge to the limits a and b. Conversely, if these two 
sequences tend to the limits a and 6, the ^-sequence tends to the 
limit a+ib. 

THEOREM. The necessary and sufficient condition that the 
sequence should have a limit is that, corresponding to any e, 
an n can be found such that z n+p z n \ < e, where p is any 
positive integer. 

This condition is necessary, for, if I be the limit, 



It is also sufficient, for it involves the conditions 



which determine the convergence of the x and y sequences. 

Uniform Convergence of a Sequence. It may be that all the z's 
are functions of a variable f : this is indicated by writing z n (g) 
for z n . Then if, at all points f in a given region, the sequence 
is convergent and has the limit (), and if an in can be found 
such that, for all points within the region, ^n(D"~^(DI < e 
when n^m, the sequence is said to converge uniformly within 
that region. 

24. Curvilinear Integrals. Before defining definite inte^ 
grals of functions of a complex variable, we shall define curvi- 
linear integrals, and prove Green's Theorem. 



23, 24] 



CURVILINEAR INTEGRALS 



43 



Consider a curve C (Fig. 24) joining two points A and B in 
the (x, y) plane. This curve can be divided into segments AL, 
LM, MN, . . . , such that, for each of these segments, only one 
value of y corresponds to each value of x ; and thus in each 
segment y is a uniform continuous function of x. Denote these 
functions by ^(x), <f> z (x), $%(%), 

Now let f(x, y) be a uniform continuous function of x and y 
in a region of the plane containing the path C. Then the 




FIG. 24. 



functions f{x, ^(x)}, f{x, <j> 2 (x)}, f{x, </> B (x)}, ..., are uniform 
continuous functions of x on the arcs AL, LM, MN, ... , respec- 
tively, and the integrals 

fl Cm Cn 

f{x, ^(x)} dx, f{x, </> 2 (x)} dx, f{x, </> B (x)} dx, . . . , 
a J I J m 

where a, I, in, ... t b, are the abscissae of A, L, M, . . . , B, are 
ordinary definite integrals. They are the Curvilinear Integrals 

J A M y) dx, J ^ f(x, y) dx, jjffa y)dx, ... , 
and their sum is the Curvilinear Integral 

\ f(x, y)dx. 
Jc 

Similarly, by dividing C into segments in each of which x is 
a uniform function of y, we can define the curvilinear integral 

I \lr(x, y) dy. By combining these a third type of curvilinear 
is obtained. 



44 FUNCTIONS OF A COMPLEX VARIABLE [CH. iv 

COROLLARY 1. If x and y are uniform functions of a para- 
meter t, the integral becomes 



where t Q and t t correspond to the initial and final points of C. 

COROLLARY 2. If x and y are uniform functions of f and q, 
and if the curve T in the ( rf) plane corresponds to the curve C 
in the (x, y) plane, 



COROLLARY 3. If C be divided into n segments by points 
(x^ y^\ (x 2 , y 2 ), ... , (x n+1 ,y n+1 ), taken in order on the curve, where 
(a?!, T/J) and (aj n+1 , 2/ n +i) are the points A and B, and if (, i^), 
(2* %)' (f> tfnX are points taken at random on these seg- 
ments, the sum 



tends to the limit f {/(aj, y)dx+\js(x, y)dy} when the law of 

Jc 

division is made to vary in such a way that n tends to infinity 
and the greatest of the segments tends to zero. 

COROLLARY 4. 

f {f(v,y)dx + ^(x ) y)dy}=-[ {f(x, y)dx + ^(x, y)dy}. 

J BA J AB 

COROLLARY 5. If K is any point on C, 

f {f(x, y)dx+\f,(x, y)dy} = \ {f(x, y)da+^(x, y)dy} 

J AB J AK 

>, y)dx+\Is(x, y)dy}. 



COROLLARY 6. If C is a closed curve, the value of the integral 
is independent of the position of the initial point, but its sign 
depends on the direction in which the curve is described. 

Differentiation under the Integral Sign. If f(x, y, a) and 

- f(x, y, a) are continuous functions of x and y on the curve C, 

eta. 



24, 25] 



GREEN'S THEOREM 



45 



and of the real parameter oc between assigned limits for oc, and if 



1 
J 



= 1 f(x, y, CL)dx, then <J>(OL) has a derivative given by 
c 



For 



= {/(# y> a+Ao.) f(x, y, a.)}dx. 
Jc 

Now, for points on C, f(x, y, oc) is a function of two variables 
x and oc ; hence ( 13) 

f(x, y, oc + Aoc)=/(a, y , oc) + |^/(^, 2/, aj 
where A tends uniformly to zero with Aoc. 
Therefore 



Aoc, 



and the latter expression tends to zero with Aoc. 
Thus 0(oc) has a derivative, given by 



25. Green's Theorem. This theorem gives an important 
relation between a double integral and a curvilinear integral. 

Let the functions P(#, y) and Q(&, y) be uniform and con- 
tinuous, and possess continuous partial derivatives, in a simply- 




.V 

Fio. 25. 



X X 



connected region containing a closed curve C. Consider the 



taken over the simply-connected area enclosed by G. 

Assume in the first place that C (Fig. 25) is a curve such that 



46 



FUNCTIONS OF A COMPLEX VARIABLE 



[CH. IV 



no line parallel to either of the axes cuts it in more than two 
points. 

Let y l and y 2 (y% ^ y { ) be the values of y on C corresponding 
to any value of x, and let A and B be the points on C of 
minimum and maximum abscissae x and X . 



Then ~ If dxdy= ~ft 7(x ' &>- p <*' 

The latter expression is the sum of the two curvilinear integrals 
-f P(x,y)dx, \ P(x,y)dx; 

J AQB J APB 

and therefore, since I P(x, y) dx =\ P(x, y) dx, 

J AQB J BQA 

~ J J ^ dX dy = J c P (X ' y) dX) 
the integral being taken round C in the positive direction. 

Similarly J J ^ dx dy = \ Q (x, y) dy. 

Hence Green's Theorem, 



holds for the region considered. 

Next, if C does not satisfy the condition that no line parallel 
to either of the axes cuts it in more than two points, the region 
can be divided into regions each of which possesses this property. 
For example, if in Fig. 26 the points A and B, at which the 



YJ 



o 




FIG. 26. 



tangents are parallel to the y-axis, are joined by a straight 
line, the two regions so obtained are of the type required. 



25] GREEN'S THEOREM 47 

Hence 

f((-^ + ! Q )<fo%=( (Pdx + Qdy)+\ (Pdx + Qdy) 
JJV ay ox/ JAQBA JABPA 

= { (Pdx + Qdy),. 
Jc 

since the sum of the integrals along AB and BA is zero. 

Thus the theorem can be shewn to hold for all simply-connected 
regions bounded by closed curves. 

COROLLARY. The area of the region enclosed by C is given by 
any of the three integrals 



Multiply -Connected Regions. Consider the region between 
the curves C and C' (Fig. 27). This region can be made simply- 




FIG. 27. 

connected by drawing a line LM from C to C'. Hence 






= \ 

J 



+ Qdy), 

G J C' 

where the latter integral is taken positively round C'. 

Similarly, for the region between the curve C (Fig. 28) and 
the ?i curves c lt c 2 , c a , ..., c n , it can be shewn that 



48 FUNCTIONS OF A COMPLEX VARIABLE [OH. 

Example. If Pd.v + Q,dy is a complete differential, she\y that 

D 



wlrere C is a closed curve. 




, I 






FIG. 28. 



26. Definite Integrals. Let f(z) = u(x,y) + iv(x,y) be a 
uniform continuous function of z in a given region, and let ACB 
(Fig. 29) be a curve in this region connecting the points z and z. 




FIG. 29. 



Let z lt z z , ..., z n , be n points taken in order on this line, where 
z n is z, and let f x , f 2 , ..., f n be arbitrary points on the segments 



SUm 



26] DEFINITE INTEGRALS 49 

the real and imaginary parts be separated, we obtain 



where = 

Now if the law of division of the curve ACB varies so that 
n tends to infinity and the greatest of the segments tends to 
zero, this latter expression tends to the limit ( 24), 

I (u dx v dy) + i \ (vdx + udy), 
J ACB J ACB 

or I (u+iv)(dx + idy). 

JACB 

This limiting value of S n is called the integral of the function 
f(z) taken along the curve ACB, and is written 

JACB 

COROLLARY 1. From the theory of limits it follows that, 
corresponding to any e, an n can be found such that 

COROLLARY 2. f f(z)dz= - { f(z)dz. 

J BCA J ACB 

COROLLARY 3. [ f(z)dz~\- \ f(z)dz= [ f(z)dz. 
JAC JC'B JACB 

COROLLARY 4. {fi(z)+A( z ) + ~-+f n (z)}dz 

= f A(z)dz+ [ /()(&+...+[ f(z)dz. 

JACB JACB JACB 

COROLLARY 5. I kf(z)dz=1c\ f(z)dz, where A: is a constant. 
JACB JACB 

COROLLARY 6. I f(z)dz= \ /{0(f)} <}>'() d, where 

J ACB J ay ft 



s a 

holomorphic function of f, and the path ACB in the 2-plane 
corresponds to the path ocy/3 in the f-plane. 

M.F. D 



50 FUNCTIONS *OF A COMPLEX VARIABLE [OH. iv 

For f(z)dz = I (u dx v dy) -f- i \(vdx + u dy) 

J ACB J J 



since z is a holomorphic function of 




COROLLARY 7. The modulus of the integral is finite. 

For, let M be the greatest value of \f(sm on ACB ; then, since 



where I is the length of ACB. 

COROLLARY 8. If F(z) is a holomorph c function whose 
derivative is f(z\ \f(z)dz = F(z)-F(z ). 

Jz 

For, let F(z) = U(x, y)+iV(x, y) ; then / 






Therefore 

I f(z)dz = \(u dx v dy) -f 'i \(v dx + udy) 

K3U 




Now the integrands are complete differentials; therefore the 
integrals are the limits of the sum of the increments of U(#, y) 



26, 27] CAUCHY'S INTEGRAL THEOREM 51 

and V(aj, y) obtained in going along ACB from (oj , y ) to (a;, y). 
Hence 



Since F(z) is single-valued, it follows that the value of the 
integral is independent of the path. 

Example. Shew that I z n dz=(z n+l -ZQ H+l )/(n + l) for all integral values 

J*9 

of n except - 1. If n is negative, the path must not pass through the origin. 
See also Examples IV. 1-4. 

Consider now the integral I f(z)dz, where the path C goes to 
-/". Jc 

inimity. 

By means of the transformation z c = l/f, where c does not 
lie on C, C is transformed into a finite path C' with f=0 as 
final point, and the integral becomes 



In order that the integrand /(c + l/f)/f 2 should be continuous, 
Lim/(c-f l/f)/f 2 or ~Limz 2 f(z) must be finite. Hence the given 

f >0 z < 

integral has a definite value if Lim z 2 f(z) is finite. 

2 >! 

^Example. I -7= - J ^C = l> provided that the path in the z-plane does 
not go through the origin. 

27. Cauchy's Integral Theorem. If a function f(z) is holo- 
morphic in a simply-connected region A, and if C is a closed 

contour lying entirely within A, I f(z)dz = 0. 

Jo 

'Letf(z) = u + iv; then, by Green's Theorem, 

ff(z)dz=\ (udx vdy) + i\ (vdx + udy) 
c Jc Jc 



the double integrals being taken over the area enclosed by C. 
Hence (equations (A), 15) 



52 FUNCTIONS OF A COMPLEX VARIABLE [OH. iv 

Example. From the integral / - , where C denotes the circle \z =1, 

deduce JcZ ' 

l+2cos0 T/1 



The following important theorems are corollaries of Cauchy's 
Theorem : 

THEOREM 1. Let f(z) be holomorphic within a simply- 
connected region, and let the paths ABC (Fig. 30) and ADC 




2-0 A 



FIG. 30. 



joining the points z and z lie entirely within the region. Then 

Jf(z) dz + I f(z) dz = 0\ 
ABC J CDA 

so that f(z) dz = \ f(z) dz. 

JABC JADC 

The integral is therefore independent of the path, so long as 
the path lies entirely within the region. 

THEOREM 2. Under the conditions of Theorem 1,F (z) = \ f(z)dz 
is a holomorphic function of z. 

Let the increment AF(0) of F(0) correspond to the increment 
A0 of z ; then 

fz+Az fz fz+Az 

AF() = /(*) dz - f(z) dz = /(f) df 

J ZG J ZQ ^ s 

|*z+Az fz+Az 



Iz+Az 



Now take A0 so small that l f()f(z) \ <e for all points f on 
the line joining z and z + Az ; then 



Therefore 



<. 



27] CAUCHY'S INTEGRAL THEOREM 53 

Hence F(z) has a definite derivative f(z) ; it is therefore 
holomorphic throughout the region. 

From this theorem the method of Partial Integration can be 
derived exactly as for the real variable. 

As in the theory of integrals of real functions, we say that a 

function F(z). which is such that ^-^ssf(z) t is an Indefinite 
Integral of f(z) ; and we write 



Example. Prove I log z dz = z log z z. 

THEOREM 3. Let/(z) be holomorphic in* the ring-space bounded 
by the curves C and C' (Fig. 27); then 



the integration in both cases being in the positive (or negative) 
direction. 

For, by Cauchy's Theorem, 

f f(z) dz + f f(z) dz - \ f(z) dz + [ f(z) dz = 0. 
Jc JLM Jc' JML 

But f /()<&= -f fa)dt. 

J ML J LM 

Therefore f f(z) dz = { f(z) dz. 

Jc Jc' 

Similarly, if f(z) is holomorphic in* the region between C 
(Fig. 28) and the n curves c lt c 2 , ..., c n , it can be shewn that 



THEOREM 4. If a is a point enclosed by a curve C, 

tdz 
= 27Tl. 
c z a 

Round a describe a small circle c of radius r ; then (Theorem 3) 
f dz _f dz 

Jc^'Jc^V 

* Here, as in Cauchy's Theorem, it is to be understood that the boundaries lie 
inside a region in which /(z) is holomorphic. 



54 FUNCTIONS OF A COMPLEX VARIABLE 

On c let z a = re ie ; then dz = re ie i dO. Therefore 



[CH. IV 



G z-a 

28. Cauchy's Theorem : Alternative Proof .* The following 
proof of Cauchy's Theorem does not depend on Green's Theorem. 

The proof will be taken in three parts : firstly, for C a 
triangle ; secondly, for C an arbitrary polygon ; and, lastly, for 
C any closed curve. 

(1) Let C be a triangle A (ABC in Fig. 31), and let the mid- 
points D, E, and F of the sides be joined, so that the triangle is 
divided into four congruent triangles A', A", A 7 ", A iv . 




D 

FIG. 81. 



Now integrate round these four triangles in succession in the 
same (positive) direction, as indicated by the arrows. The two 
integrals along each of the lines DE, EF, and FD cancel each 
other, so that the net result is the integral round A in the positive 
direction. Hence 

- [ /(*)<fo=f /(*)<**+[ f(z)dz+\ f(z)dz+\ f(z)dz; 
JA JA' JA" JA'" J A lv 



so that 



f(z)dz 



f 

J 



z)az 



There must therefore be at least one of these smaller triangles 
we denote it by A x such that 

f(z)dz 



Cf. Knopp, Funktionentheorie, Vol. I. 



27, 28] 



CAUCHY'S INTEGRAL THEOREM 



55 



Dealing with A a in the same way, we obtain a triangle A 2 
such that 



[f(z)dz ^4 \f(z)d 

J A! J A 2 

\f(z)dz 

J A 



and therefore 



Proceeding thus, we obtain a sequence of similar triangles 
A, A!, A 2 , ..., each contained by the preceding one, and such that 



i-A n , and 



\f(z)dz ^4 J 



As n tends to infinity, the triangle A n shrinks to a point, f say, 
which lies within every one of the triangles A, A x , A 2 , 
Now, corresponding to any e, an y can be found such that 

| X |< c if z f | < >7, where 



Let ti be chosen so great that A n lies entirely within the 
circle | z f | = v\ ; then 



f f(z)dz= \ f(Qdz+ f f 

JA JA JA 



The first two of thes integrals vanish ( 26, Corollary 8), 
so that ., ., 

f(z)dz=\ \(z-)dz. 

J An J A,, 

Therefore If f(z)dz ^f \\\\z-\\dz\ 

I J A u J A,, 

O 

= -jc8 n , where s n is the perimeter of A n , 



Hence 
Therefore 



e / -s 1 \' 2 
. } ( ^j , where s is the perimeter of A. 



f f(z)dz = 0. 
JA 



56 FUNCTIONS OF A COMPLEX VARIABLE [CH. iv 

(2) If C (Fig. 32) is a quadrilateral, it can be divided into two 
triangles A and A' by a diagonal which lies within it ; then 



f f(z)dz=\ f(z)dz+\ f(z)dz = 0. 
Jc JA JA' 

Similarly, if C is any polygon, it can be divided into triangles 




FIG. 32. 



by diagonals lying within it, and, since the integrals along these 

diagonals cancel each other, I f(z) dz = 0. 

Jc 

(3) Let C be any closed curve ; then, as in 26, 

< n 

f(z)dz = LimS n> where S n = ^f(z r )(z r -z r . l ). 
JG r=i 

Now let f(z) =f(z r ) 4- r] r f or points z on the straight line joining 
z r -i and z r , and let the law of division of C vary so that, for 
r=l, 2, 3, ..., n, |^ r |<e/(2L), where L is the length of C, 
and also 



Then, if P is the polygon of vertices , z lt z 2 , ... , z n , 

\ /(*> <fo = s r </w + *> dz = s + s r *** 

JP r = \Jz r -i r = lJ2 r _! 



But f(z)dz = Q; therefore 
Jp 



11 



Therefore 

Jo" 

Hence 



28, 29] C^JCHY'S RESIDUE THEOREM 57 

29. Cauchy's Residue Theorem. If the point a is the only 
singularity of f(z) contained in a closed contour C, and if 



V 



has a value, that value is called the Residue of f(z) at a. 

From Theorem 3, 27, it follows that, if encloses several 
singularities, the sum of the residues at these points is 



The following cases are important : 

CASE 1. If n is any integer except 1, the residue of (0 a)~ n 
at a is zero. 

CASE 2. The residue of (z a)~ l at a is unity. .' 

CASE 3. If 

f(z) = &J(z - a) + A 2 /(z - a) 2 + . . . + A n /(z - a) + 0(), 
where 0(2) is holomorphic at a, the residue of f(z) at a is A r 
Example. Shew that the residue of (2s + 3)/(s I) 2 at 1 is 2. 

CASE 4. If f(z) is holomorphic at a, the residue of f(z)/(z a) 
at a is /(a) : for 



i ( f( Z ) _ i r /(a) i r 



X 

c 



Now take C so small that for all points on it X <e; then 



where i is the length of C. 

Hence -Uf -dz=f(a). 

ZtnJQZa 

This is equivalent to the following theorem : 

THEOREM 1. If Lim {(z-a)f(z)} is a definite number A, the 

z >a 

residue of f(z) at a is A. 



58 FUNCTIONS OF A COMPLEX VARIABLE [OH. iv 

It follows that, if <f>(z) and \fs(z) are polynomials in z, and if 
z a is a non-repeated factor of \Js(z\ the residue of <f>(z)/\ls(z) 
at a is <f>(a)l\l/(a). 

Example. Shew that the residues of (pz 2 + qz + ?)/{ (z - a) (z - b) } at a and b 
are ( pa 2 + qa + r)/(a - b) and (pb 2 + qb + r)/(b - a). 
See also Examples IV. 5-7. 

Multiply-Connected Regions. If f(z) satisfies the conditions 
of Theorem 3, 27, except for isolated singularities at points 
in the space between C and the curves c l} c z , ..., c n , the sum 
of the residues at these points is 



Residue at Infinity. If f(z) has an isolated singularity at 
infinity, and if C is a large circle which encloses all the singu- 
larities of f(z) except infinity, the residue of f(z) at infinity is 
defined to be 



taken round C in the negative direction (negative with respect 
to the origin), provided that this integral has a definite value. 

If the transformation 2=l/f be applied to the integral, it 
becomes 



taken positively round a small circle about the origin. Hence 
it follows that, if Lim { /(!/)/} or ^^ m { ~ z f( z )} nas a definite 



value, that value is the residue of f(z) at infinity. 

Example. Shew that the residues of z/{(z-a)(z-b)} and (z 3 - 
at infinity are - 1 and 1. 

Xote. Both of these examples shew that, if a function is holomorphic at 
infinity, it does not necessarily follow that its residue there is zero. 

THEOREM 2. If a uniform function has only a finite number of 
singularities, the sum of the residues at these singularities, that 
at infinity being included, is zero. 

Let C be a closed contour enclosing all the singularities of 
f(z) except infinity: then the sum of the residues at these 
singularities is I 



29, 30] CONTOUR INTEGRALS 

But the residue at infinity is 



59 



Hence the sum of the residues is zero. 

Example. Evaluate the residues of z*/{(z- l)(z- 2)(2-3)} at 1, 2, 3, and 
infinity, and shew that their sum is zero. Ans. 1/2, -8, 27/2, -6. 

See also Examples IV. 8-10. 

30. Evaluation of Definite Integrals. Many definite in- 
tegrals can be evaluated by means of integrals round closed 
contours. 

Example 1. Prove T cos x d * = , where >0. 

Jo x't + a' 1 2a 

Integrate f(z) = e* z /(z* + a 2 ) round the contour (Fig. 33) consisting of : 

(1) the A'-axis from R to R, where R is large ; 

(2) that half of the circle ] z \ = R which lies above the #-axis. 




O R X 

Fio. 33. 



The only pole of f(z) within the contour is ia, at which the residue is 



Li m \(z-ia 

z-^ia \ 



e ~ a 
55" 



Hence / /(.,-) 
But 



^c 1 
P 






TrR 



Hence the integral along the semi-circle tends to zero aa R tends to 

infinity. 

Therefore 
so that 



f" - S>Y dx- v *~* 
Jo x* + a* 2a ' 



60 FUNCTIONS OF A COMPLEX VARIABLE [CH. iv 

THEOREM 1. Let AB (Fig. 34) be that arc of the circle z\ = E 
for which 0^0^ 2 , where = amp3; and let zf(z), as R tends 




FIG. 34. 



to infinity, tend uniformly to the limit K, where K is a constant ; 
then f 



Lim 



JAB 



For, let zf(z) = K + X, and choose R so great that X | < e ; then 
f K + X . IP 2 

= \ 9 \dO 
<e(0 2 -0 1 ). 
Hence Lim f(z) dz = i(0 0,)K 

R->< J^ 



AB 



For example, in Example 1, | zf(z) \ ^ R/(R 2 - a 2 ), so that K = 0. 
Example 2. If m > 0, prove 



T 

Jo 



cos mx dx TT - V m ; 



[Integrate e imz /(l+z 2 + z*) round the contour of Fig. 33.] 

From Theorem 1 it follows that, if f(z) = (j>(z)/ifs(z), where 
\fs(z) is a polynomial of degree n, and 0(0) is a polynomial of 
degree less than n, 

Limf f(z)dz = i(0^0 l )^, 
R^^JAB o 

where a and b are the coefficients of z 11 ' 1 and 2 n in 0(2) and 
\fs(z) respectively. 

In particular, if the degree of </>(z) is ^n 2, a is zero, and 
therefore the integral of f(z) round the contour of Fig. 33 gives 



r: 



30] WORKED EXAMPLES 61 

where 2 denotes the sum of the residues of f(z) at points above 
the #-axis. 

Example 3. Prove / / x = v ' , where a > 0. 
Jo XT + a* 4er 

The residue of l/(2 4 + 4 ) at a pole a. is 



But the poles above the #-axis are ae ln /* and ae l37r / 4 . 



Therefore |_ _^_ i = 2 M A e-' : 



Hence / -^-.-^. 



The inequality sin^^20/7r, where 0^0^7r/2, is frequently 
found useful. 

Example 4. Prove J - 2 ' ^ dr = ^ e~ a , where a > 0. 

Integrate /(2)=ze <z /(,2 2 + a 2 ) round the contour of Fig. 33. The only pole 
within the contour is m, the residue at which is e~ a /2. Hence 

f m 

J-K /W 

But r f(Re)Reide ^ I *T^"'7 dO 



2R 2 

2 

2 - 2 



a ./ o 



JR 2 ri 
^R 2 



2R 2 f 
2 -a 2 Jo 



TrR 

Hence Lim [V(Re ie )R^'^<9=0. 

Thei'efore / y 

- 

so that r^ sil 

>'o ^ 2 4 

>. Integrate e'* over the following contour (Fig. 35) : 




O R X 

Fin. ;;;.. 



(1) the .f-axis from <) to I! ; 



62 



FUNCTIONS OF A COMPLEX VARIABLE [CH. iv 



(2) the circle |2| = R, from 0=0 to #=<x, where . j ; 

(3) the line #=., from |s| = K to O. 

When R tends to infinity (1) gives I e~ x<i dx or . 

/ ^ 

On (2)e-z 2 =e- R2 cos20 e -iR2sin20 ; so that, if 20=7r/2-<, 



*dz 



= e ~ & si d<, where /? = Tr/2 - 2o. 



Accordingly, when R tends to infinity, this integral tends to zero. 
Again, on (3)z=re la , and therefore, when R tends to infinity, thejjntegral 

becomes r 00 

- e ~ ' C08 2a { cos(r 2 sin 2cx) - i sin (r 2 sin 2a.) } e la -dr. 
Jo 

But the integral is holomorphic within the contour ; hence 



/" e- 

J(\ 



E. (cos rx - 1 sin (/.). 



Therefore, if the real and imaginary parts are equated, 

I e~ x " cos 2<x cos(# 2 sin 2o.) dx ^- cos a., 
Jo 2 

and / ~ x2cos2a sin(.r 2 sin2a.)o?^=-^sina.. 

Jo 2i 

If a.=7T/4, these integrals become the Fresnel Integrals 

r 2 r ^^ 

I cos SV* d\v ~^ \ sin '?'*"' ct-& == ~ A 
Jo Jo 4 

Example 6. Prove / --J = -r-^ , where < a < 1. 

Integrate f(z) = e az /(I+e z ) round the contour (Fig. 36) consisting of the 





Y, 


27TZ 




1 _ 


"It 


-R 


O 

Fie 


~R 

. 36. 



.r-axis and the lines x R, y = 27r. The only pole within the contour is TTZ, 
at which the residue is - e a7ri . 



30] CONTOUR INTEGRALS 63 

If *=R + z)/, then \f(z) \ ^ e nR /(e* - 1 ), so that Lim/(*)=0; hence the 

R >oo 

integral along #=R vanishes when R tends to infinity. 

If z=-R + iy, then | fts) ^ e -" R /(l -e~ R ), so tha,t Lim /"()= 0; thus the 

R->oo 

integral along x= R vanishes when R tends to infinity. 

1 f 

Therefore ^j 

Hence 

Therefore 



The transformation 6^=^ changes this integral into 



l+y sm TT 

Two methods can be employed to evaluate integrals of the 
type I /(cos 0, sin 0) clO, when f(x, y) is rational in x and y. 

J -7T 

The first is to use the transformation # = tan ^0. The integral 

{+00 
R,(x)dx, where R(a?) is 

rational in x. 

The alternative method is to apply the transformation z = e ie , 
and integrate round the circle \z = 1 . 

Example 7. Prove / - ^ ; = -^"~ , , where the sign of *J( 2 -b z ) 
J a + bcosti ^/(*-6*) 

is chosen to satisfy the inequality | a - *J(a' 2 - b 2 ) \ < \ b \ ; it is assumed that 
a/6 is not a real number such that - 1 =a/b^. 1. 



!*= 

' 



d dz dz 



- 

ib J c (z - 



- 

Jo + 6cos(9 i c bz* + 2az + b ib c (z - OL)(Z - / 
where C is the circle \z\ =1, and a. and /? are the roots of 
Since ./? = !, it follows that either |o. or \/3\ is less than 1, or that 
1 a. | == | ft I = 1. The latter alternative is excluded, however, since in that case 
a/6 would be real and such that -l^a/6^1. Let a.=(-a + Ja*-b'*) /6, 
where the sign selected for \/ 2 - b' 2 is that which makes | a. | < 1. Then 

THEOREM II. If Lim(z a)f(z) = K, where A: is a constant, 

fz->a 
f(z)dz, the integral being taken round the arc 

^O., of the circle z a\ r, is i^ O^K. 



64 



FUNCTIONS OF A COMPLEX VARIABLE [OH. iv 



For, corresponding to any e, an r\ can be found such that 
if | z a < rj, A | < e, where (z a)f(z) = K + A. Hence, since 



Therefore 



Prove 



Lim (f(z) dz = i(0 2 - 
/- >o J 

f'iiEf^^f. 

Jo # 2 



Integrate f(z) = e iz /z round the contour (Fig. 37) consisting of 

(1) the ,^-axis from r to E, where r is small and R large 

(2) the upper half of the circle | z \ = R ; 

(3) the .r-axis from R to r ; 

(4) the upper half of the circle \z\ = r. 




-R 



R X 



FIG. 37 



Let I be the integral duetto (2) ; then 



=f< 



iRcosfl E sinfl n i 



Hence 



e de 



Therefore Lim I = Q. 

R-o 

Again, Lim2/(2)=l, so that (Theorem II.) the integral along (4) tends to 
ITT as r tends to zero. 

Hence Lim | J e -dx+ ^ ^-ctoj=7n. 

Therefore 



\\ T hen, in the description of a contour, part of a small circle is 
described to avoid a singularity of the integrand, the contour 



30] PRINCIPAL VALUE 65 

is said to be 'indented' at the singularity: for example, the 
contour of Fig. 37 is the contour of Fig. 33 indented at O. 



Example^. If 0<v< 1, prove / - dx=- 



o 

= 2 p ~" 1 /(l +z) round the contour (Fig. 38) consisting of : 
(1) the ^-axis from r to R ; (2) the large circle | z | = R ; 
(3) the .?;-axifi from R to r ; (4) the small circle | z \ T. 




FIG. 38. 

Within this contour f(z) is uniform. Consider that branch for which 
amp 2=0 on (1). 

Since p >0, Lini2/(2) = : hence the value of the integral along (4) tends 

*-*o 
to zero as r tends to zero. 

Again, when \z = R, 2/(2)|^R p /(R 1) : therefore, since jo<l, the 
integral along (2) tends to zero as R tends to infinity. 

At the point 1 amp 2 = 77: hence the residue at this point is eCf--*)**. 
Also on (3) amp ,s = 27r. Therefore 



r^B-l 
cfe*ftrttfP- 1 K 
1 -\-x 

Hence f^^^' . 

Jo \+x sinpTr 

The substitution x = e y transforms this integral into 



Principal Value of an Integral. If /(z) is holomorphic in a 
region containing that part of the ic-axis for which a = x = b, 
except for a simple pole at a point c on the #-axis, where 
a < c < 6, then 

" 



tl-nds to a definite limit as e tends to zero. 



66 FUNCTIONS OF A COMPLEX VARIABLE [CH. iv 

r& ^7/v, 



= log (b c) log (c a). 
cfce 



T . 
Hence Lim 



Now, let/(s) = 0(2)/(2-c); then ( 15, Theorem, p. 29) 



where X is continuous in the region. Therefore 
Lim { f f(x)dx + i f(x)dx\ 

e->0 Ua Jc+e J 



This limit is called the Principal Value of I f(x)dx, and is 
ritten - 6 

P /(aj)do;. 

Ja 

Example 10. If < a < 1, prove 

rx a ~ i 
- dX = 7T COt 7T. 
1-.^ 

Integrate z?~ l /(z-l) round the contour of Fig. 38 indented at 1 (Fig. 39). 




Example 11. If TT < a < TT, prove 

rsinh ax 7 1 a 
-^ dx -^ tan - 
smh Tr.r 2 2 

Integrate e a2 /sinh (TTS) round the rectangle (Fig. 40) of sides ?/ = 0, y = l t 
x= R, indented at O and' 2. 



30,31] THEOREM ON RESIDUES 67 

Example 12. Integrate e ibz l(r-\-iz) a , where 0< < 1, r >0, b >0, round 






O 

FIG. 40. 



R X 



the contour of Fig. 41, where it is assumed that amp (r+ iz) is zero at points 
on the i/-axis between and ir ; and thus prove 

- 4* 27T ,_,_, r 



r; 



Prove also 



f + * *'** 



=0 J and shew that 




o 

PIG 41. 



If r= 1, .r = tan ^, deduce 

F (cos 6) a -' 2 cos ad cos (6 tan 6}dd 




'*T(a) ( 



31. Theorem. Let C be a closed curve such that f(z) is holo- 
inorphic within and on C and 0(0) is meromorphic within and has 
no singularities or zeros qn C ; then 



where n 1 , a.,, a :i , ... are the zeros of 0(0) w r ithin C of orders 
r i r 2' r s-" respectively, and 6 1? 6 2 , 6 3 ,... are the poles of <j>(z) 
within C of orders s p *.,, ,s.,, ... respectively. 



68 FUNCTIONS OF A COMPLEX VARIABLE [CH. rv 

For (p(z) = (z a l ) ri \Is(z), where \fs(z) is holomorphic at a x ; hence 



so that 



<p(z) z-a, 
The residue of the integrand at a x is therefore 



Similarly, since (2? 6 1 )* 1 ^() = x( X where x(^) i- s holomorphic 
at b lt the residue at 6 X is 

Hence 

OOBOLLA.T1. 
OOBOLLABY 2. 



Example 1. If ^>(^) is a polynomial of degree , shew that 2^ = ?^. 

Example 2. If <^(*) is a polynomial with factors 0., z /3, ..., shew that 
oc~* : +)8~ A: +...= -B, where /: is any positive non-zero integer, and R is the 

residue of -i$fe) a ts = 0. 
*^() 

32. Liouville's Theorem. A function which is holomorphic 
at all points of the plane, including infinity, must be a constant. 

Let f(z) be such a function; then, if a and b are any two 
distinct points, the only singularities of the function 



are a and &, and possibly infinity. But since LimzF(2) = 0, the 

.-> ce 

residue of F(z) at infinity is zero ( 29, p. 58). Now the sum 
of all the residues is zero ( 29, Theorem 2) : hence 



so that /(a) =/(&) ; and therefore, since a and 6 are arbitrary 
points, f(z) is a constant. 

COROLLARY. Every function which is not a constant must 
have at least one singularity. 



31-34] DIFFERENTIATION UNDER INTEGRAL SIGN 69 

33. The Fundamental Theorem of Algebra. If f(z) is a 
polynomial in z, the equation /(z) = has a root. 

For, if not, the function I/f(z) would be finite and holomorphic 
for all values of z, and would therefore be a constant (Liouville's 
Theorem). Hence f(z) would be a constant, which contradicts 
our hypothesis. 

34. Differentiation under the Integral Sign. Let the func- 
tion f(z, f) of the two independent complex variables z and f be 
holomorphic with regard to both z and f so long as z lies in a 
region A of the z-plane and in a region A' of the f-plane. Then 

the function #().= I f(z, )dz, where C lies entirely in A, is holo- 

f 7^ 

morphic at all points of A', and 0'(f ) = I ^/(z, f ) dz. 

Let f(z, ) = u-\-iv and 0() = P-f- / iQ, 

so that P = I (udx v dy), Q = I (vdx + u dy) ; 

then ( 24), 

3P_f Cdu, 'dv_, \ 9P_f fdu 
3?~JcW 3? y )' ^~Jr 



3Q f /9v 7 , 9^ 7 \ 3Q f 

^l = V^^ + ^^2/J' ^ = 

Sf Jc^f 3 y ^ J 

Hence (equations (A), 15), 



Thus 0(f) is a holomorphic function of f : its derivative is 
given by 



JExample. Integration under the Integral Sign. Shew that, if C' and 
C' lie in A and A' respectively, 

( ff(z,C)dzd{=[ i f(z,t)d{dz. 

Jc f Jc Jc Jc f 

Let {" and f be the lower and upper extremities of C'; then/ / f(z,t)ddz 
is holomorphic in , ;ind J ^ c ' 



70 FUNCTIONS OF A COMPLEX VARIABLE [OH. iv 

Hence ( </>(K= ( I /(*, C)d(dz-[ f I f(z, {}d{dzl 

JC' JG JC' I 'C -'C' Ib = bO 

= f f /(z, 

Jc ./c' 



35. Derivatives of a Holomorphic Function. A function 
f(z) which is holomorphic in a simply-connected region enclosed 
by a curve C, possesses derivatives of all orders at every point 
interior to C. 

For, if z is any point interior to C, 



Now let A be a region which contains the point 0, and whose 
boundary is interior to C. Then the f unction /()/( 2) is holo- 
morphic with regard to both f and so long as f remains on C 
and z in A. Hence ( 34), 



Similarly, by means of repeated differentiations, it can be 
shewn that 



COROLLARY 1. If C is a circle of centre z and radius R, and M 
is the maximum value of \f(z) \ on C, |/M()| 



COROLLARY 2. If f(z) is continuous at all points of a finite 
('not necessarily closed) path C, the function 



is holomorphic in 2 at all points which do not lie on C, and its 
n derivative is 



COROLLARY 3. If ^(a;, y)+iv(x t y) is a holomorphic function 
of z = x + iy, then t6(aj, y) and v(o5, 2/) have partial derivatives 
of all orders. 



35] DERIVATIVES OF A HOLOMORPHIC FUNCTION 71 

EXAMPLES IV. 

1. Prove / = log , a cross-cut being taken along the negative real 
axis. J * z z o 

2. Under the same restriction as in the previous example, prove 



where n may have any value except 1, and the same branch of z n is taken 
on both sides of the equation. 

3. Prove \*e az dz = (e -!)/. 



f 

4. Prove / cos az dz = sin (az)la. 

Jo 

5. Shew that the residue of e az /(l + e z ) at -i is - e a7ri . 

6. If K is any integer, shew that the residue of cot z at KTT is 1. 

7. Shew that >\% residues of e zi /(z 2 + a' 2 ) and ze zi /(z 2 + a 2 ) at ai are e~ a /2ai 
and e~ a /2 respectively. 

8. Shew that the sum of the residues of any rational function is zero. 

9. If /(,)= A_ + _^* + ... + ,- f " , shew that the residue of f(z)l(z-x) 
at a is /(.') 

10. If /"(2) = 2A r /(2-a) r + <(2), where ^(2) is holomorphic near a, shew 

] n 

that the residue of /()/(*-*) at a is -2 A r /(^-a) r . 

11. Shew that, if m and n are positive integers, and m<.n, 



M 



12. Integrate ze imz /(z 4 + a 4 ), where m and a are positive, round the contour 
of Fig. 33, and shew that 



ma 

= ^<f^sin"!-. 



IQ T>^^, 
13. Prove 



rCQ&ma; , JT . /7a TT\ 
/ _ ^6;= - ^e V 2sm( + T 

Jo ^ 4 + 4 2 3 V/2 4/ 



14. Integrate e iz /(z-ai), where >0, over the contour of Fig. 33, and 

shew that {+* a COB x + x sin x , 

( L: 

J-a, .C 2 + (l 2 



15. By integrating e iz l(z + ai\ where a>0, prove 
acoBx+xmnx 7 



ifi 



p,., , ^ in ll1 -' 7 TT ~rr ma 

L *+& <f -'' = 2 e C08 ' 



where 



72 FUNCTIONS OF A COMPLEX VARIABLE [CH. 

17. If < a < 2, shew that 

/27T. + 7T\ 

(i) r af^dx 2?r C S \ 6 
U Jo 



sin (**) 



f o lx+x* tJ3 sin^ra 
[Integrate - - round the contour of Fig. 37, and equate real and 
imaginary parts.] 

Prove j[ i -^_ _ -^_- 2 ^ c ?^ = ^log(l4-r), if -l<r<l 

f-), if r<-l or r>l. 



[Integrate ^ ^g^ 1 ~J!> rou nd the contour of Fig. 33, and 

put ^7= tan ^.] 

19. If > 0, and - Tr/2 < ^ < 7r/2, prove 

and 

[Integrate s"-^- 2 round the contour consisting of the positive .?>axis, the 
line amp2=#, and part of an infinite circle.] 

20. If ^0, prove 

x.x /"^(l-f.r^cosa^ , TT _y? a 
l) --^ = e 2 COS 2 ; 



/;;\ r* ^s 
J. 1+ 



[Integrate e tez /(l'+2 + 2 ) round the contour of Fig. 33.] 

21. Integrate e~ z ~ round the rectangle of sides y = 0, 2/ = , x= E, and 
show that /+ /+ 

/ e -(-+w> 2 ^ ^ e -W.r = N /TT. 

J-X J-OD 



Deduce: 




= \7r, where c is any constant. 



22. Integrate e iz l(z + a\ where >0, round the square whose sides are 
,r = 0, # = B, y=0, ^ = R, and shew that : 



ii) r <fe =rj^i ( fa. 

Jo ^ + Jo l + ,r-' 



rv] EXAMPLES IV 73 

23. Integrate e~ j2 round the rectangle whose sides are .^ = 0, .v = R, y = 0, 
?/ = >, where 6>0, and shew that : 



' (ii) [%-* sin 2&r dx = e - 6 ' 2 / * 
Jo /o 



24. Integrate e nz /cosh TTZ round the rectangle of sides .r= K, v/ = 0, y = 
and shew that 

r cosher 7 1 a 
dx = - sec = , wne re 
coshw.r 2 2' 

f2r ^Q- a -bi J/} = 27TI, if 6 > 0, 

25. Prove cot c7^ .' ... 7 ' 

Jo 2 = - 2?ri, if 6 < 0. 

r2ir 

26. Prove I cos"^o?^ = 0, if n is odd, and 



27. Prove that, if y is the unit of circular measure : 

k T +x sin x , IT , 



r+ ao 1 - cos x 

I1) L T^-i 



28. If a is positive, prove 

TT . ,,.v /"" .r sin aa; , TT 

=-2 Smm ' ; (u)P /o ^-^^ = 2 



29. If r and 6 are positive, and < a < 2, prove 



[Integrate (i) zf t ~ l e ibz /(z 2 + r^) round the contour of Fig. 37, and 

(ii) z a ~ l e ibz /(z z -r 2 ) round this contour indented at r and -r.] 



30. If -l<a<l, prove 



Deduce that, if -!<<!, 

I ^ " /"^ 

Jo Jo 

[Integrate J.A '", ^ ~ , '* round the contour of Fig. 37, and put 



- < >S ArtTT 



74 FUNCTIONS OF A COMPLEX VARIABLE [CH. 

31. Prove 7fi /1-r.V'. 

-1 



[Integrate _ round the contour of Fig. 37.] 

32. If >0, prove f^D^ Jl (1 _,-). 

Jo .v(x 2 + a 2 ) 2a 2V 

33. If a > 0, prove 



_ a 

[Integrate -^ - ^-r round the contour of Fie:. 37.1 

2(logZ- ITT/2) 

34. If a > 0, prove 

{ 1 - cos (a tan 0) } + log (cos (9) sin (a tan 6>) 

J_ ~ 



f ~ (log cos By 2 + <9 2 

1 _ giaz 

^ - r-r round the contou 
z\og(l-iz) 

35. If b>0, r>Q and 0<a<2, shew that 



1 _ giaz 

[Integrate ^ - r-r round the contour of Fig. 37.1 
z\og(l-iz) 



[Integrate s a - 1 e ei6 7(^ 2 + ^ 2 ) round the contour of Fig 37.] 
36. If 0<a<2, prove / ^_^ 2 ^ 



Deduce ( 

.... f .^- 1 -^- 1 dx A Tra / 7r6 

(11) / , = log tan , / tan - 

x Jo log^? \-\-x 1 \ 4 / 4 

where 0<6<2. 

37. Let P(-s) and Q(z) be polynomials of degree m and n respectively, 
where m^n 2, and let Q(z) have no positive or zero real roots. By means 
of the integral of P(z)~Logz/Q,(z) taken round the contour of Fig. 38, prove 

r^Mdv- R 

Jo Q(^)^ 

where H denotes the sum of the residues of P(^)Log2;/Q(2:) (0 < amp s < 2?r) 
at the zeros of Q(z). 

38. By integrating (Log^) 2 /(l+^ 2 ) round the contour of Fig. 38, prove 

floga? , 
I ,^0^ = 0. 
Jo l+.r 2 

39. By integrating log(4-t)/(2 2 +l) round the contour of Fig. 33, prove 



Deduce 



iv] EXAMPLES IV 75 

40. Prove / --^ dj; = 7r r - 

J smh.t' 4 



41. If a is real, show that 

sinew?, TT . , aw 



42. If i > - 1, w > 1, and m - n is an even positive integer, prove 

j" 00 sin n# sin 7i.# , _ e~' 1 e~ m 
J "(l + # 2 )sin# r e-tf- 1 

r (logo?) 2 , 16 

43. Prove / 2 ^ = QT~^S 7r - 

Jo 1+,P+^ 2 81v/3 

[Integrate (LogzY/(I + z + z 2 ) round the contour of Fig. 38.] 

44. If Kjt?<l and - TT < A < TT, shew that 

p jy-^dla? TT sin;? A 

Jo 1+2^ cos A + x- ~ sin pir sin A 

45 ' 



[Integrate zl(u-e- iz ) round the rectangle of sides 0;= TT, y=0, y = 
46. If r>0, s>0, 0<a<l, 0<6<1, a + 6>l, shew that 

<b ' +X dx 



9 ^ a. M-- 



_ 

~ 



Deduce //(cos ^) ^(fe 

47. By integrating e z ' 22 /s round a suitable contour, shew that 



= <fo=f . 

x 4 

Deduce / dx**^* 

Jo ^ 2 

48. By integrating e' : /\/* along a suitable path, shew that 

rcos.g . _ /""sin x 
V/# - *J X 

49. If < . < 7T/2, shew that 

/"-t- 00 t&n~ l xdx TTOL 



[Integrate log(l -^)/(5 2 -22sino.+ l) round the contour of Fig. 33.] 

50. Integrate ^"/(e 2 ' 2 -!), where a is real, round the rectangle of sides 
= 0, .i?=R, y = 0, .y=l, indented atO and i, and shew that 

/" sin'/.v . 1 , /a\ 1 

Jo ^ -\' f ''' = T ' 



[CH. V 



CHAPTER V. 

CONVERGENCE OF SERIES: TAYLOR'S AND LAURENT'S 

SERIES. 

36.* Convergence of Series. Let S w denote the sum of the 

cc 

first n terms of the infinite series 2 w n> where the w's are real 

51=1 

or complex quantities ; then, if S n tends to a finite limit S as n 
tends to infinity, the series is said to converge or to be convergent 
and to Have the sum S. The necessary and sufficient condition 
for this is ( 23) that a number m can be found such thai, when 
n^im and p is any positive integer, 

S n +p &n i <C Or j W n+l + W n+2 + . . . 4- W n +p \ <C * 

Ii w n = u n + iv n , the series Stt n and ^v n converge to the 

real values U and V, where U + iV = S; for I ^u n ~U 

I i 
V are both less than |S n S|. Conversely, if the 

I 

series Su n and 2/y w converge to the values U and V, the series 
iv n ) will converge to the value U + iV, since 



Absolute Convergence. If the series of moduli ^ I w 

71=1 

convergent, the series Zw n is also convergent, since 



a series of this kind is said to be Absolutely Convergent. The 
series *Lu n and 2v n are then also absolutely convergent, since 



*In this and the following paragraphs some definitions and theorems on 
infinite series which will be found useful in the course of this work are 
summarised ; for fuller proofs and for further information on the subject 
reference may be made to Bromwich's Theory of Infinite Series. 






36] THE HYPERGEOMETRIC SERIES 77 



\v n \ = \Wn\- Conversely, if 2w n an d 2)v n are 
absolutely convergent, 2iv }l will be absolutely convergent, since 



.AWe. The value of an absolutely convergent series is inde- 
pendent of the arrangement of the terms.* 

Multiplication of Series. Since 

(u n 4- iv n )(u' m 4 iv' m ) = u n u m - v n v' m 4 iu n v' m 4 iv n u' m , 
the product of the two absolutely convergent series 1/w n and 
Sw' n is equivalent to 

Zw n 2u' m 2-Vn 2/v' TO 4 i S^ n Zt/ m 4- * 2v w 2u' m . 
Hence the product is the absolutely convergent series 

W^ W\ 4 (Wj -H/2 4- W z W\) 4 (tVj t(/ 3 4 ^2 ^'2 4 ^3 ^'i ) 4 . . . . 

Most of the series with which we shall have to deal will be 
absolutely convergent series. The tests for convergence of series 
of positive terms apply also to absolutely convergent series : the 
most important of these is : 

GO 

The Ratio Test. If Lim w n+l /w n \< 1, the series Vw 7l is 

,l->co i* 

absolutely convergent: if Lim [t0 n+1 /w ll |>l j the series is 
divergent. 

If Lim iw n+1 /w n \ = l, further tests must be applied: one such 
/i > /> 

testt is the following: 



If 



IV, 



n+1 



n n 



where /u, is a constant and | co n \ is less than a fixed number A 
for all values of ??, the series 2| w m ' is convergent if /m ]> 1 and 
divergent if /x ^ 1. 

1. Shew that the Hypergeometric Series 



1'snlutely convergent if \z\<l and is divergent if \z\>l ; while, if 
| s | = 1, it converges absolutely if E (y - rx - (3) > 0. 

*Cf. Unnnwich. 75. tCf. Bromwich, 12, 79. 



78 FUNCTIONS OF A COMPLEX VARIABLE [CH. v 

Example 2. If R (y -.-/?)> 0, prove 

F(, ft y, l) = ( y ( -^_-^F(, ft y + 1, 1). 
Let T w denote the nth term of F(o., ft y, 1) ; then, if n = 1, 2, 3, ... , 



T _T ^ 

" 1 



y/ 1.8...(y+l)(y+8)...(y+) 



_Yl /A T ' rp" 

\ y/ M+1 ~ M + 2 ' 

where T w ' and T" are the w th terms of F(., /3, y + 1, 1) and F(.- 1, ft y, 1) 
respectively. Also 



Hence, since LimT,, = 0, 

" yF(o.-l, ft y, l) = (y-F(o, ft y + 1, 1). 
Again, if w = l, 2, 3, ... , 



so that (y-o.-/?)F(a,fty, l) = (y-o.)F(cx.-],fty, 1). 

Hence F(o, ft y, l) = (y ,"' ) F(, ft y + 1, 1). 



Example 3. Shew that, if the series ^,w n is absolutely convergent, the- 

00 1 

series Zlog(l +w n ) is also absolutely convergent. 
Choose n so large that | w n \ < 1 : then 



Hence an m can be found such that, for n^w, 



where C is a constant independent of n. 

Therefore, if "2f | w n | < c, "z" | log (1 + w) | < Ce ; 

m m 

so that the series Zlog(l +w n ) is absolutely convergent. 

37. Convergence of a Double Series. If ^ and o>. 2 are com- 
plex quantities such that CD.,/^ is not real, the double series 

+ CO +00 1 

SY 1 
^ 



is absolutely convergent. The accent indicates that the term 
for which m = ti = is omitted. It is convenient to assume 
I(ft> 2 /ft> 1 )>0 : if this is not the case, interchange w l and w 2 . 



36, 37] CONVERGENCE OF A DOUBLE SERIES 



79 



Divide up the plane (Fig. 42) by parallel and equidistant lines 
into parallelograms similar and equal to parallelogram OABC, 
where A, B, and C are the points 2^, 2w 1 + 2o> 2 , and 2<o 2 . Since 
the angle AOC lies between and TT. One term of 




FIG. 42. 



the series corresponds to each angular point of the net-work, 
except the origin. 

Consider those angular points which lie on the parallelogram 
PQRS, the mid-points of whose sides are 2pto ly 2pw<,, 
where p is a positive integer. There are 2p + I points on 
each of the sides, and therefore, since the four vertices 
eacli lie on two sides, there are Sp angular points on the 
parallelogram. 

Xo\v let i/ be the shorter of the two perpendiculars from O on 
AB and BC. Then for each of the angular points on PQRS 

1 _ 1 



so that 



8 



wheiv tlir summation rxtnuls to all the points on PQRS. 



80 FUNCTIONS OF A COMPLEX VARIABLE [CH. v 

Now, if the values 1, 2, 3, ... , be assigned to p in turn, all the 
angular points in the plane will be included. Hence 

ri 3< ^ 3 \r 2+ 22 + 3* +< 



and therefore the series is convergent. 

00 

38. Power Series. Let ^c n (z a) n be a power series, and 

7t = 

let the ratio c n /c n+1 tend to R as n tends to infinity. Then 
from the Ratio Test it follows that the series is absolutely con- 
vergent within and is divergent without the circle | z a \ = R. 
This circle is called the Circle of Convergence and R the Radius 
of Convergence. 

Example. Shew that the radius of convergence of the geometric series 
+ z 3 + ... is unity. 



At a point on the circle of convergence the series may or may 
not be convergent. A test for absolute convergence is given in 
36. The following test is sometimes useful when the series is 
not absolutely convergent. 

Abel's Test. If the coefficients c 1 , c 2 , c 3 , ... , form a decreasing 
sequence of positive numbers, c n tending to zero as n tends to 

00 

infinity, the sum 2 c nZ n converges at all points of the unit circle 
i 

except possibly at z= 1. 
For, consider the series 

qcos + c 2 cos 20+c 8 cos 30+ . .. , < < ZTT. 



Let S mi _p= 

m+l 

and let s r = c 

so that cos 

cos (m + 2) = 2 s l , 



Then S m> P = c m+l 8 1 + c m+z (s. 2 8 1 )+ ... J f-c m+p (s p -s p . l ) 

Si (tfjH+l ~~ c m+->) H~ *-2( c m+-2 ~ Cm 
i Sp - 1 V^m+p - 1 



Now s r = sin(ir0)cos { 
so that -I/sin J0^s r ^ I/sin J0 . (r = l, 2, 3, ...). 



38] POWER SERIES 81 

Therefore, since all the quantities 

^m+i Cin+2> ^m+2 ^m+3> > ^m+p > 

are positive, 
S m , p = ^ m+1 ~~ 



nnrl Q > m+l 

^^-sinp' 

But, by making in large enough, c m+1 can be made arbitrarily 
small. Therefore, since < W < TT, the series is convergent. 
Similarly, since 



the series (^sin #+c 2 sin 20+c 3 sin30+... , can be proved con- 
vergent if < 6 < 27T. Hence the series 



sn ^ 

converges if 0<$<27r. 

This theorem can be illustrated as follows : 

If amp z =^= TITT (n integral), the terms of the series can be 



(a) 




(b) 



O A 2 A 3 A, X 
O A, A 2 A 3 > X 

FIG. 43(o)(6)(c). 

represented by OA 1? AjA.,, A 2 A 3 , ..., {Fig. 43 (a)}, where each line 
makes the same angle amp z with the preceding one. These lines 

M.F. F 



82 FUNCTIONS OF A COMPLEX VARIABLE [OH. v 

form a kind of spiral, and A n tends to a point, which represents 
the sum of the series. If amp z TT the lines will be alternately 
positive and negative {Fig. 43(6)} and the series will be convergent; 
but when amp = {Fig. 43 (c)} the method does not apply. 

Example. Shew that z + z 2 / < 2,+z?/3+ ... converges for \z\ = l except at 
2=1 ; and deduce that the series 

cos W cos W 
COS0 + ^ + ^ + .-, 

a , sin W sin 3(9 , 
sm0 + ^ + 3 +..., 

are convergent if = S^TT. 

Multiplication of Power Series. If the two series 

2^ w and 240" 
o o 

are convergent within the circle | z = R, their product 

c o c 'o + ( c o c 'i + c i c/ o> + ( c o c/ 2 + c i c 'i + C 2 c/ o)^ 2 + - 
is also convergent within that circle (cf. 36). 

39. Taylor's Series. Let f(z) be holomorphic in the region 
bounded by a circle C of centre a and radius R, and let z be any 
point within C such that | z a r < R : then 




n\ 

Now, since | f ^R r for all points f on C, it follows 
( 26, Cor. 7) that 

M 



where M is the maximum value of |/(f ) | on C. But this quantity 
can be made arbitrarily small by increasing n : hence 



38, 39] TAYLOR'S SERIES 83 

for all points within C. This is Cauchy's extension of Taylor's 
Theorem.* 

The convergence is absolute, for (35, Cor. 1) the modulus of 
each term is not greater than the modulus of the corresponding 
term of the absolutely convergent series 



Let z be the nearest singularity to a : then if z be any point 
within the circle of centre a and radius | z l a \ , R can be chosen 
so that z a <R<|0 1 a\. Thus the Taylor's Series converges 
absolutely at z, and therefore its radius of convergence is | z l a \: 
that is, the circle of convergence of the Taylor's Series is the 
domain of the point a. 

COROLLARY 1. If f(z) and its first TI 1 derivatives vanish 
at a, while f (n) (a) is not zero, a is a zero of f(z) of order n. 

For example, z = kir is a zero of sin 2 (17): this zero is a simple zero 
since cos*, the derivative of sin 2, is not zero at the point. 

COROLLARY 2. If f(z) and (j>(z), and also their first n-1 
derivatives, vanish at a, while < n > 



Example. Prove 

z-*-0 

COROLLARY 3. If /<*>( a ) = (71 = 0, 1, 2, ...), f(z) vanishes 
identically at all points in the domain of a. 

Example 1. Shew that, for all points within the circle | z | = 1, 



and deduce that j log(l+s)|^-log(l - z\). 

Example 2. Prove / Iog(sin7r.r)cfa;= -log 2. 

Integrate log (sin TT*) round the rectangle of sides .r = 0, .r=l, ^ = 0, y = R, 
imputed at and 1. 

The integrals round the small quadrants at and 1 vanish in the limit ; 
hence / log(sin7TA-)c?^ 

= il [log (sin 7T/y)- log {sin (Tr + Triy)}]dy + I log { si n (TTV 

jo 

*Cf. 43, Note. 



84 



FUNCTIONS OF A COMPLEX VARIABLE [OH.V 



Now, since w = sin ITZ = sin TTX cosh Try + i cos TTX sinh Try, 
as x increases from to 1, (#>0), w passes round the curve PQR (Fig. 44), 




FIG. 44. 

from P(i sinh Try} when ^ = to Q(cosh Try) when # = 1/2, and to R( - i sinh Try) 
when x = \ : hence amp (sin TTZ) decreases by TT, so that 
log (sin Triy) log { sin(7r + Triy)} = TTI. 

Again si 
Therefore 



Hence J log(sin irx)dx= - log 2 + J log(l - e Zwxi - 2irR )dx. 
But T log(l _ e 2^-2 ff B)^. < -log(l -e-^ R ), 

which tends to zero as R tends to infinity. Therefore 

P 

/ log (sin TTX)dx= log 2. 

Jo 

Example 3. If | z \ < 1, prove 

(i) tan~^ = z - 03/3 + ^ 5 / 5 r (Gregory's Series) 



where the principal value of tan" 1 ^ is taken in each case. 

JGL 




FIG. 45. 



40. Laurent's Series. Let f(z) be holomorphic in the ring- 
space bounded by two concentric circles C x and C 2 (Fig. 45) of 



39, 40] 



LAURENT'S SERIES 



85 



centre a and radii R t and R 2 , (R 1 <R 2 ). Then if z is any point 
within the ring-space, so that 



f(z) can be expanded in a series of the form 



J_f ./Cf) 

Wc^ 



Now let Mj_ and M 2 be the maximum values of \f(z)\ on 
and C 2 ; then ( 26, Cor. 7), since \z a\=r,- 

,n+l 



l-r/R 2 VR 2 / 



^r/Rj-1 \r 

But these two quantities can each be made as small as we 
please by increasing n ; hence 



where 



A - - 

- 



1. Since 



and 



and R 1 <|2 a -<R 2> it follows that the series is absolutely 
convergent for all points within the ring-space. 

* Cf. 43, p. 95, Note. 



86 FUNCTIONS OF A COMPLEX VARIABLE [OH. v 

Note 2. Since f(z) is holomorphic between C^ and C 2 , the 
integrals round these contours can be replaced by integrals round 
any concentric circle C of radius R, such that R^R^Rg. It 

+ GC 

follows that f(z) = 2 A^(0 a)^, where 

p - co 



Note 3. Let <j>(z; a) and ty(z\ a) represent the series 

00 CO 

^A p (z a) p and ^A_ P (3 d)~ p respectively. 
o i 

Then /()$(*; a)+\fs(z', a\ where 0(0; a) is holomorphic 
within the circle |0 a| = R 2 , and \K0; a) outside the circle 

|0-a| = R r 

Principal Part at a Pole. If the only singularity within 
| z a R x is at a, Rj can be made arbitrarily small. Then if 

n 

\/s(z; a) = 2-A-_ p (0 a)~ p > where n is finite, f(z) has a pole of 

fml 

order TI at a, and \/r(0 ; a) is called the Principal Part at the 
pole. If ifr(z m , a) is an infinite series, f(z) has an essential 
singularity ( 22) at a. 

Example 1. If f(z) is holomorphic in the region bounded by a closed 
curve C except at the poles a lt 2 > a > an( i ^ G r {l/(z-a r )\ is the principal 
part of f(z) at a r (r= 1, 2, ... , ), shew that 



where f is any point interior to C. [Cf. Exs. IV., 9.] 

Example 2. If | z \ > 1 , and the principal value of tan" 1 z is taken, shew that 



according as 

41. Fourier Series. A uniform function F(z) which satisfies 
the equation F(0-fQ) = F(2;) for all values of 0, where Q is a 
non-zero real or complex number, is said to be a Periodic 
Function, and to have the period Q. It follows that, if ra is 
any integer, positive or negative, F(0-j- ?nf2) = F(0). If no integer 
2>(j?=/=l) can be found such that Q/p is a period of F(0), Q is 
called a Primitive Period of the function. A function which 
has only one Primitive Period is said to be Simply -Periodic. 



40, 41] 



FOURIER SERIES 



87 



Now let the function f(z) have the period 2&>, and let =e inz/<a . 
To each value of f corresponds an infinite number of values of 
z, differing by multiples of 2o>. Therefore to each value of f 
corresponds one and only one value of f(z), so that f(z) is a 
uniform function of f. 

Let A (Fig. 46) be the point 2&>, and let R denote an infinite 
region of the z-plane, bounded by two lines parallel to OA, 




FIG. 46. 

in which f(z) is holomorphic. Now if z is any point on a line 
through z l parallel to OA, z = z l + \u>, where X is real, and there- 
fore ^ e inz il^e iir ^, so that |f | is constant. Hence such a line is 
represented in the f-plane by a circle with the origin as centre, 
and as z increases by 2o>, passes round the circle once in the 
positive direction. Any portion of the region R bounded by 
two straight lines perpendicular to OA, and at a distance OA 
from each other, is therefore represented on the f-plane by 
a ring-space bounded by concentric circles with the origin as 
centre. 

In this ring-space f(z) is holomorphic since 

'*_ df(z) 
if'-* * ' 
Hence, by Laurent's Theorem, 

+ 00 +00 



where A, = 

C being any circle in the ring-space with the origin as centre. 



88 FUNCTIONS OF A COMPLEX VARIABLE [CH. v 

Therefore 



where a p = - [ 2 "f(z)cos^dz, and 6.= - 
o)Jo' o> co 

This is Fourier's well-known expansion : it is valid for all 
points within the region R. The function /(z), it must be noted, 
is holomorphic in R. 

42. Classification of Uniform Functions. Functions which 
are holomorphic for all finite values of z are called Integral 
Functions. Such functions are developable by Taylor's Series 
throughout the plane. From Liouville's Theorem it follows that 
every integral function which is not a constant must have a 
singularity at infinity. 

THEOREM 1. An Integral Function for which infinity is a 
pole of order n is a polynomial of degree n. 

For, if f(z) be such a function, then by Laurent's Theorem 



where 0(f) is holomorphic at f=0. Hence 



Therefore <j> (I/z)=f(z)- (B 1 s + B 2 z 2 + ... + B n n ). 

Accordingly 0(1/0) is holomorphic for all finite values of z. 
Hence, since 0(1/0) is holomorphic at infinity, it must, by 
Liouville's Theorem, be a constant, B say. 

Therefore f(z) = B + B^ + B 2 2 + . . . + E n z n . 

Polynomials are also known as Rational Integral Functions. 

An integral function which is not a polynomial is called a 
Transcendental Integral Function. The Taylor's Series contains 
an infinite number of terms, and thus the function has an essential 
singularity at infinity. Examples of such functions are e z , cos z, 
and sin z. 

An integral function f(z) which has no zeros in the finite part 
of the plane can be put in the form e G(z \ where G(z) is integral. 
For the function G (0)3= log {f(z)} has no singularities in the 



41, 42] INTEGRAL AND RATIONAL FUNCTIONS 89 

finite part of the plane, and is therefore an integral function : 

hence f(z) = e G(z) . For example, e z has no zeros except at infinity. 

The ratio of two polynomials is called a Rational Function. 

THEOREM 2. If f(z) is meromorphic throughout the plane, 
and if infinity is either an ordinary point or a pole, f(z) is a 
Rational Function. 

Let there be m poles 04, 2 , ..., a m , in the finite part of the 
plane ( 22, Th. 2, Cor.), and let the principal part of f(z) at a r be 
<t> r (z) = AfVC* - Or) + A ( 2 r) /(s - a,) 2 + . . . + A/(Z - a r )*v, 

(r=l,2, ...,m). 

7)1 

Then /(2)~#r(3) is finite at all finite points of the plane. 



Accordingly, since ^(z), < 2 ( )> > $(2)> are all zero at infinity, 
m 

must be a constant or a polynomial, say 



Hence /(0) = 0r(^) + V^( )' which is a Rational Function. 



COROLLAEY. A meromorphic function other than a Rational 
Function must have an essential singularity at infinity. 

EXAMPLES V. 

1. Shew that the series : 



are absolutely convergent for all values of z. 

oo oo 

2. Shew that the series ^c n z n and the series of derivatives y,nc n z n ~ l 

o i 

have the same radius of convergence. 



3. Shew that the radius of convergence of the series ^nlz 11 is zero. 
[Such series do not define functions.] 

4. Shew that the product of the series f t z n /n ! and f>"Y?i ! is 2(*+ z') n ln\ . 

00 

oo z n+l 

5. Shew that the series S / . ^ is absolutely convergent at all points 
on its circle of convergence. ^ $ 



90 FUNCTIONS OF A COMPLEX VARIABLE [en. 

6. Shew that, for all finite values of z : 



(ii) cose ^l- + i...; 



(iii) sma =z- - + --...; 
(iv) cosh* = 1+^ + ^ + .... 
7. Shew that, for all values of ??, the Binomial Theorem, 



..., 

holds for all points within the circle |s| = l, that branch of (l + z) n being 
taken which has the value unity when z = 0. 

8. If the function /(*) has an essential singularity at a, shew that l/f(z) 
has also an essential singularity at a. 

9. Shew that the series 



is convergent if R (-&)>. 1/2, and find its sum. Ans. I+z. 

10. Prove that, if\z\<l, 

T . 6:r 



11. Prove 

l-coss 1 /.-XT- s -sin 2 1 



12. Prove that, if R(s)> - 1, 



13. Prove that, if | s | < 1, 

l{log(l+2)}2 

14. Shew that the series 



converges if | z \ < 1, and that its sum is z/(l - z). 

15. Shew that the series 

z z* z* z* 

is convergent if |s|<l and also if |s|>l, and that the respective sums are 
*/(!-*) and !/(!-*). 

16. Shew that the series 2 q^e?" converges for all finite values of z 



v] EXAMPLES V 91 

17. Shew that, with the notation of 37, the series 



is absolutely convergent if A> 2. 

18. Shew that the series 

a . 2?n + 2 2 / ...- 

1 ! 2 ! % ! 

where w is a positive integer, is absolutely convergent if \z\ <m m /(m + l) m+1 . 

19. Shew that the radius of convergence of the series 



20. Shew that the series 



2 ! 3 ! 

is convergent if | z \ < 1/4. 

21. If a > 0, shew that 



[Integrate (**'- 1 -i2 + ^ 2 /2)/{2 3 (a 3 + 2 2 )} round the contour of Fig. 33.] 
cos .^ 2 + sin o^l 7 



rtrt -TV / 

22. Prove / 

Jo 



o x 

[Integrate (e**- l)/z 2 round the contour consisting of the positive x and y 
axes and a quadrant of an infinite circle.] 

23. If a and b are positive, prove 



30 

24. Shew that, if a and m are positive, 

f a 



[CH. VI 



CHAPTER VI. 

UNIFORMLY CONVERGENT SERIES: INFINITE 
PRODUCTS. 

43. Uniformly Convergent Series. Let S n (z) denote the 

GO 

sum of the first n terms of the infinite series 2 w n ( z )> whose 

n = l 

terms are functions of z ; then if, at all points of a region A, the 
sequence S^z), S 2 (0), S 3 (0), ..., converges uniformly (23), the 
series is said to be Uniformly Convergent in A. The necessary 
and sufficient condition for this is that, corresponding to any e,* 
an m can be found such that, for all points of A, 



where p = l, 2, 3,..., and n^m. The region A is a closed 
region ; i.e., the points on the boundary are included. 

Example. If the series 2 ^n(z) converges uniformly in a region A, and if 

n=l , 

f(z) is finite in A, shew that the series 2f(z)w n (z) converges uniformly in A. 
In the following three theorems it is assumed that the series 



oo 

s 

n 



W, 



> n (z) is uniformly convergent in the region A. 



THEOREM I. If Wi(z), w 2 (z), w 3 (0), ..., are continuous in A, 

CO 

the function S(z) = ^w n (z) is also continuous in A. 

n = l 

For, if z and z + kz are points of A, an m can be found such that 



where n^m. But, since S n (z) is continuous, an 17 can be found 
such that, for | Az | < 17, 



* It should be noted that e is independent of z. 



43] UNIFORMLY CONVERGENT SERIES 

Hence, if | Az < >?, 



93 



z) - S n (z) | 



<. 



Therefore S(z) is continuous in A. 

CO f 

THEOREM 2. The series ^ 1 w n(z)dz, where C is a path i 

.ic j. 

the region A, is convergent and has the sum I S(z)dz. 

For, since at all points of A 



= f {S(z)-$ n (z)}d 
IJo . 



<el. 



where is the length of C. 

COROLLARY. If the initial and final points of C are Z Q and 0, 

S(z)dz, Wi(z)dz, w z (z)dz, ..., 

Jzo Jz Jz 

CO /3 

are functions of z, and 2 I *p w (*)dfe converges uniformly in A, 

n = l Jz 

since a maximum value can be assigned to I. Accordingly, if a 
uniformly convergent series be integrated term by term, the 
resultant series is also uniformly convergent. 

THEOREM 3. If w^z), w 2 (z), w 3 (z), ..., are holomorphic in A, 
S(z) is holomorphic at all interior points of A, and 



Let f be any interior point of A, and let C be the boundary of 
a simply-connected portion of A of which f is an interior point. 

\n+ 

Then if, for all points of C, 



^ 



w n (z)(z- 



n+] 

i+p 



wliere cZ is the shortest distance from f to C, and 7c + 1 >0. Thus 



94 FUNCTIONS OF A COMPLEX VARIABLE [CH. vi 



)~ k ~ l dz converges to the sum 

J 



c c 

and therefore, since 



In particular, if fc 



Now, this integral is holomorphic (35, Corollary 2) at 
Accordingly S(f ) is holomorphic at and has derivatives given by 

Ad*u>() ,, 



COROLLARY. If C (Fig. 47) is the boundary of a simply- 
connected portion of A, and if C' is the boundary of a region A" 
interior to C, the series of functions of f , 



will be uniformly convergent in A', provided d > 0, where d is 
the shortest distance between C' and C. 




FIG. 47. 



Example. If the terms of the series S(z) = ^w n (z) are holomorphic in the 

region contained by a closed contour C, and if the series converges uniformly 
on C, prove that 8(2) is holomorphic within C. 



S 

a v nd uniformly convergent in the region A, provided that a con- 



Weierstrass's M Test. The series 2jW n (z) will be absolutely 

i 



43,44] WEIERSTRASS'S M TEST 95 

CO 

vergent series of positive constants ^ M can be found such that, 

i 
for all points z in A, |w n (|^M ri , (71 = !, 2, 3, ...). 

For, if M n+ i 4- M n+2 + . . . + M n+p < 6, 



. Since the moduli of the terms of the series 



employed in the proof of Laurent's Theorem ( 40), are Jess than 
the corresponding terms of the series S( r /^W^ anc ^ x](^i/^) n > 
the series integrated are uniformly convergent on the paths of 
integration. Thus the consideration of the remainder can be 
omitted from the proof, provided that the M Test has been 
previously proved. The proof of Taylor's Theorem ( 39) can 
then be contracted in a similar manner. 

00 

Kxample 1. Shew that the circle of convergence of the series ^z n /n 2 is a 
region of uniform convergence. 

00 

Example 2. Shew that the series SVC 2 " - w 2 ^ 2 ) represents a meromorphic 

function with poles at the points TT, 77, 877, ____ 

Let z be any point of the region bounded by |s| = R, where 



Then j * mr \ ^ mr R, where n = m + l, m + 2, ... ; and therefore 



1 



1 



Accordingly, since the series 2)l/(ir B) 1 is convergent, 2 l/(2 2 wV 2 ) 
converges uniformly at all points of the region. 

TO 

Now the function 2 l/(s 2 -wV 2 ) is holomorphic at all points of the region 

except the poles TT, 27r, ... , WITT. Hence the given series is holomorphic 
in the circle except at these points. But R can be chosen so large that any 
assigned point lies in the circle ; therefore the series is holomorphic at all 
points except TT, 27r, 877, ____ 

44. Power Series. Let R be the radius of convergence of the 

QO 

power series 2 c n (z a) n . Then if R^R, the area of the circle 



I : n = R x is a region of uniform convergence. 

For, corresponding to any e, an in can be found such that 
^m, 

! V+i- < t ( P = 1 , 2, 3, . . . ). 



2 



96 FUNCTIONS OF A COMPLEX VARIABLE [CH. vi 

Therefore, if z a =R 1? 

c n+l (z-a) n + 1 + c n+2 (z-a) n + 2 +...+e n+p (z-a) 

^ | c n+1 1 V+ 1 + | c n+2 1 V+ 2 + . . . + c +1 , | R 1 +* 

Since any point within the circle of convergence can be 
enclosed in a region of uniform convergence bounded by 
| z a =R-p where | f a <R 1 <R, it follows that the series 
gives a holomorphic function at all interior points of the circle of 
convergence. 

+ CO 

COROLLARY 1. If the series 2 c n(z a) n converges for 

CO 

; a I < R 2 , it will be uniformly convergent for 

Example. If f(z) is defined by the series ^c n (z-a} n ^ convergent for 
< | z a | < R, shew that the residue of f(z) at a is c^ . 

COROLLARY 2. If f(z) is holomorphic and has the Laurent 

00 

Expansion ^ A p z p in the region R^I^I^Ra, and if infinity 

00 

/is the only singularity exterior to 0|=R 2 , the residue of f(z) 
at infinity is A_ 1 . 

Example 1. Prove that the residues of e l i z at the origin and at infinity are 
+ 1 and - 1 respectively. 

Example 2. If n is integral and ^ 1, prove that the residue at infinity of 
that branch of * " which is positive when z is real and > 1 is 

/1.3.6...(2n-3) 1.3. 5. ..(2 -5) +( _ 1) n-i\ 
12.4.6. ..(2w-2) 2.4.6.,.(2-4)T 

Undetermined Coefficients. Let /(z) and <(0) denote the series 



which converge in the region R x < | 2; a \ <^ R 2 , and let the 
coefficients c n , (n = 0, 1, 2, ...) be unknown. Then if 
</>(z)=f(z) for all points of tliis region, 

c n = A nj (n = 0, 1, 2, ...) ' 



>4, 45] UNDETERMINED COEFFICIENTS 97 

For, if C is the circle | - a | = R (R 1 <R< R,), 



In particular, if /(,?) = for all points in the region, 

c n = (n = 0, 1, 2, ...). 
COROLLARY. If f(z) is odd, all the coefficients of even powers 

CO 

of z in the Laurent Series f(z) = ^ c n ?l are zero ; while if f(z) 

- r s> 

is even, all the coefficients of odd powers of z are zero. 
For, if f(z) is odd, 



while, if f(z) is even, 



Example. Consider the function l/(e z -l) : it has simple poles of residue 
+ 1 at the points 0, %7ri, 4?, Hence, if < 1 z \ < 27r, 



(1) 



where the coefficients c , e,, c 2 , ... , are to be determined. 
If the sign of z be changed, 



Adding these two equations, we have 



so that c = 1/2, c 2 = c 4 = c 6 =...=0. 

Next, multiply both sides of equation (1) by e z 1 : then 



(11 
--2 



z 2 z 3 
2j + 3 1 

Hence, equating coefficients, we obtain the equations 

"'-2;2l + 3! = ' 



\\liich the coefficients c^, c 3 , c 5 , ... , can be found. 

45. Additional Contour Integrals. The calculation of resi- 
dues by means of expansions in series is found helpful in the 
evaluation of many definite integrals. 



M . F. 



98 



FUNCTIONS OF A COMPLEX VARIABLE [CH. vi 



Example 1. Prove 



where m and a are real and positive. 

Integrate f" 2 ., over the contour of Fig. 37 ( 30). When E tends to 

infinity, the integral along the large semi-circle tends to zero. When r 
tends to zero, the integral along the small semi-circle tends to - in/a*. 
To calculate the residue of the integrand at ia put z=ia + : then 

6 




Hence the residue at f=0 is - 

and therefore f */ " " 

Jo xx^ 



.fa- = -W 



from which the required result follows. 
Example 2. Evaluate f* 

Consider that branch of 
is real and > 1. 



-, where n is a positive integer. 



which is real and positive when z 



This function is uniform in the region between the great circle C (Fig. 48) 




FKJ. 48. 



and the closed contour y consisting of small circles about 1 and 1, and the 
real axis between these circles. There are simple poles at +i and - i. 
At z = i, amp(z-l) = 37r/4 and amp(z + l) = 7r/4 : therefore 



45, 46] LEGENDRE POLYNOMIALS 99 



Hecce the residue at z = i is(- l)"- 1 ^ 2 ^ 2 )- Similarly the residue at z= -i 
is -l"- 1 ^. Thus 




where the integrals along C and y are taken in the directions indicated by 
the arrows. 

But ( 44, Corollary 2, Example 2) 



.fl,3.5...(2n-3) 1 . 3 . 5 ... (2tt-5) ,/_ iy .-i 

ITi r \2.4.6...(2w-2) 2. 4. 6. ..(271-4) 






and 




^* 

\r c^-U W rT . 
46. Legendre Polynomials. Consider that branch of 



in the domain of z = which has the value + 1 when = 0. Since 
the function has singularities at f>/(f 2 1), it can (39), for 
values of z such that \z\ is less than the smaller of the two 
quantities fv/f 2 1 1, be expanded in a series 



in which the coefficients are polynomials in f. The coefficient 
P w (f ) is called Legendre s Polynomial of order n. 



Shew that 



If we expand both sides of the equation 
{l_ 2( _ f)2+8 2 } -J = {i_2f(_ 

:uul <M|uate the corresponding coefficients, we obtain 



100 



FUNCTIONS OF A COMPLEX VARIABLE [CH. VI 



Again, from the expansion for (1 



2 , it follows that 



where c is a small circle about the origin. 

Now c can be replaced by the contour of Fig. 49, described in 
the direction indicated by the arrows, where A and B are the 



ii^ 




FIG. 49. 



points \/f 2 1, C is a large circle, and y and y are small 
circles about A and B. 

The only case in which this cannot be done is when AB passes 
through O. But in order that this may be so, 



must be real and negative. Therefore, since 



the two quantities f+v/f 2 1 and f Vf 2 1 must be purely 
imaginary. 

Hence, by addition, it follows that f is either zero or 
purely imaginary. We therefore exclude the case in which f 
lies on the imaginary axis. 

The integrals along the circles C, y, and y vanish in the limit, 
while the integrals along DB and BD cancel each other : thus 

pjo-*V ds 



_ cos 



46] LEGENDRE POLYNOMIALS AS INTEGRALS 101 
where z = + \/ 2 1 cos ( 



The branch of Vf 2 1 considered does not matter, since 
cos (TT 0) = cos 0. 

The integrand has a singularity if /\/f 2 1 is real and 
numerically less than 1. In that case 2 /(f' 2 ~~-0 must be real 
and less than 1, and therefore f 2 is negative. Hence is purely 
imaginary. The imaginary axis is therefore a line of singu- 
larities for the integral. 

If f=l, P n (f) = l, so that the + sign must be taken: if 
f= 1, P n (f) = ( l) w , and therefore the sign must be taken. 
Hence, for points to the right of the imaginary axis, 



p n (f) = 
while, for points to the left of the imaginary axis, 

Again, in the equation 

P B( ) = M - _<** 

put 1/2 for 2 : then 



~2S 

since the integrand is holomorphic between C and the contour- 
made up of y, y', and AB described twice. 

Thus P n (f)= 1 (V+v/r^cos 

7T Jo 

Since P u (l) = l, we take the + sign: thus 
1 

TC Jo 

Again, let f=cos#, (0<#<7r), so that A and B become the 
points (Fig. 50) e iB and e~ ie . Then if, in replacing the path c 



102 FUNCTIONS OF A COMPLEX VARIABLE [CH. vi 



by the contour of Fig. 49, the arc AB of the unit circle is taken 
instead of the straight line AB, we obtain 



Thus, if z = 



TT J_0 N /(2cos0-2cos0) 
2 f cos( 



_ 2_ f 

~7rJ 



7rJoV(2cos0-2cos#) 

A 




In this equation let and <f> be replaced by IT and TT ; 

then 



Example 1. Prove 
(-l- 

Differentiate (1 - 



n(f) with regard to s : then 



Now multiply both sides by (1 - 2fs+2 2 ) : then 



But 



~^ 



Hence equating coefficients, we have 
(^ + l)P n+1 (0-( 

Example 2. Prove 



, -w, 1, i-Jf\ (Cf. 36, Ex. 1.) 



" 1 



(-!)!. 3... (2^- 



(l-) 2 2 



, 47] 



EXPANSION OF cotz 



103 



Therefore, equating the coefficients of z'\ we have 






1.1 



1.2.1.2 



4 



!';.'> < mple 3. From Example 2 deduce 



47. Expansion of cotz in a Series of Fractions. The 
function cot /(), where g=/=mr, has simple poles at f and 
,777, (?i = 0, 1, 2, ...): the residues at these points are cotf 
and l/(gmr) respectively. 

Now consider the integral I - - dz taken round the rectangle 

ABCD (Fig. 51) of sides x= (w + l/2)w, y=b, where n and 
b are chosen so that f lies within the rectangle. 





To each point z on AB or BC there corresponds a point -2 on 
CD or DA: therefore 

(cot sdz 



ABCDA 



where 



j'-l oofesv^sdv, I.,= | 

JAB ^"-^- J BC 



104 FUNCTIONS OF A COMPLEX VARIABLE [CH. vi 

On AB, z = x + ib ; therefore 

cot z \ = 

Hence 1 1, 1 ; 



To avoid discontinuous values of the integrand, we choose 
>r] then 



Therefore 



L H = l_ e -26 I 3.2 4. 7)2 _ r 2 T_ p -2i> ~rp o w 
/ _ co t^ |^ C' / JL ^^ C/ v/ /i w -^ /T 1 ^ 

Lim^^O. 



Again, on BC, 0= (71 + 1/2)^ + ^, so that 
cot 01 = 1 tan i 



f& 
w 



Zrdy 



Hence I 



where ^ is chosen so great that (w + )& ]> r. Thus 



i 2 |s 



Therefore 



LimI 2 = 0. 



ABCDA 



and 



a~~2 
^ 



a definite value as ?i tends to infinity 
\> 

( 43, Example 2). Accordingly, when n and 6 tend, to infinity, 
we have 



and therefore cot f = ~ 



Example 1. Shew that cosec 2 = 



Example 2. Integrate L _ round the contour of Fig. 51, and prove 



47, 48] MITTAG-LEFFLER'S THEOREM 105 

48. Mittag-Leffler's Theorem. It is possible to construct a 
function which shall be holomorphic except at isolated simple 
poles a lt a. 2 , a 3 , ... , these poles being arranged in order of ascend- 
ing moduli, provided that, for some integer n, the series 2 1/<V 1 ' 
is absolutely convergent. 

00 

Consider the series ^]w r (z\ where 



Let C be the circle z = R, where R < a p+ i \ ; then, for all 



points z in the region bounded by C, 

R 



Or 



= /u, where // = ! 



r 



and r =p + 1, p + 2, . . . . Therefore 



n , 



Hence, by Weierstrass's M Test, the series 2 U >( 2 ) converges 

P+I ' 
absolutely and uniformly in the region bounded by Q. 

Vj 

Accordingly, the series ^jW r (z) represents a function of the 

r = l 

re<[uired type in this region. But R can always be chosen so 
large that any assigned point lies in the region : hence the series 
represents a function of the required type. 

COROLLARY 1. If f(z) and 0(0) are two functions with simple 
poles of residue unity at a lt a 2 , a 3 , ... , f(z)-~ <p(z) is holomorphic 
at all finite points, and is therefore an integral function. Hence 
any function of this tyjfe-ei^be put in the form 




where G(z) is an integral function. 
COROLLARY 2. If the function 



is differentiated /> 1 times, a function is obtained with poles of 
order i> .at the points 04, ".,, ".. ..... 



106 FUNCTIONS OF A COMPLEX VARIABLE [CH. vi 

Note. These functions have all essential singularities at in- 
finity, since there is an infinite number of poles exterior to 
every circle | z j = R, ( 22, Theorem 1, Cor. 2). 

00 

Example 1.' Since the series 2 V*' 2 ^ s convergent, the function 



is holomorphic at all points except 1, 2, 3, ... , where it has simple poles. \ 
Example 2. Shew that 

1 \ 

). [Use ^ 47.] 

j 



z -K z-mr mr 

+ 00 



Weierstrass's Zeta Function. If ^SS'V^ 8 * s the absolutely 



C 

convergent series of 37, where Q = 2mt> 1 -h2?ta 2 , then, by 
Mittag-Leffler's Theorem, 



,. 



is a meromorphic function with simple poles at the angular 
points of the network of Fig. 42. This function is Weierstrasss 
Zeta Function, and is denoted by g(z), (cf. 75). 

The function is odd. For, if the order of summation is 
reversed ; i.e. if m and n are replaced by in and n ; then 



Hence -,)- -{1 + SS'(rV 5 

Weierstrass's Elliptic Function. Differentiating the Zeta Function, 
we have i + ^ 



This is Weierstrasss Elliptic Function $>(z), (cf. 72), so th; 



It is holomorphic except for poles of the second order at th< 
points 2mfc> 1 + 2'fto) 2 , where m and n take all integral valw 
Since ( z) = $(%), 1p(z) is an even function. 



48, 49] WEIERSTRASS'S ELLIPTIC FUNCTION 1 07 , 

n 

49. Infinite Products. Let P n denote the product II w r , where 
the w's are complex quantities no one of which is zero. Then if 
the sequence P I} P. 2 , P 3 , ... , tends to the non-zero limit P as n 
tends to infinity, the infinite product ILw r is said to converge to 

the limit P. 

If P is zero, the product is defined to be divergent. 

If Iiw r is convergent, Lim^ n =l; for W n =*~P n [P n . l} and P n 
and P n _! both tend to the limit P. 

THEOREM I. If the series S = y) u ' is convergent, the pro- 

i 

duct Iie w will be convergent and will have the value e s . 

For, since the exponential function is continuous, an r\ can be 
found such that, if n 

2>,-s 



e l ( 

Hence IIe w converges to e s . 

i 

Unconditional Convergence. If the series 2w n is absolutely 
convergent its value is independent of the order of summation 
of the terms, and therefore the value of the infinite product 
II'-"'" is independent of the order of the factors. When this is 
the case the infinite product is said to be Unconditionally 
'ergent. 

K.i'mnple. Shew that, if the series w,, is absolutely convergent, the 

product 11(1 + w tl ) is unconditionally convergent. 

i 
- Example 3, 36.] 

00 

THEOREM II. If the terms of the series S(z) = ^w n (z) are 

i 
holomorphic in a given region, and if the series converges 

00 

uniformly in that region, the infinite product P(s)= H> ?r " (:) will 
!>' holomorphic at all interior points of the region. 

Kr S( : ) is holomorphic at all such points ; hence P(z) = e s(z) is 
.ilso holomorphic (15, p. 30), and its logarithmic derivative i^> 

'ivi-n by PY?^ /-/S^ 

j i \ - i ' ' o v, -^ ) '^~\ / / \ 

\\:) = ~dz~ =: ^ Wn(Z) ' 



108 FUNCTIONS OF A COMPLEX VARIABLE [CH. vi 

50. Weierstrass's Theorem. It is possible to construct an 
integral function with zeros of the first order at the isolated 
points a lt a. 2 , a 3 , ..., these points being arranged in order of 
ascending moduli, provided that, for some integer n, the series 



- n i absolutely convergent. 
Let v( ,)_^ + I + J p+ ... + 5;, (,= 1,2,3,...). 

CO 

Then the series ^w r (z) converges (48) absolutely and uni- 

P+I 

formly in the region bounded by the circle |0| = R, where 
Now let 



(r =p + 1, p + 2, . . .), where the path of integration lies in the circle. 

CO 

Then the function ^W r (z) * s holomorphic in that region, and 



therefore ( 49, Theorem II) so is the infinite product II 
Hence the function 

n |Yi -~ 

1 l\ <^r 

is holomorphic in the circle, and has simple zeros at the points 

^1 > ^2 *' ^P ' 

Now R can be chosen so large that the circle includes 
any assigned point; hence the theorem holds for all finite 
points. 

Again, let f(z) be any function of the required type. Then 
if (f>(z) denotes the infinite product above, f(z)l<p(z) will be an 
integral function without zeros, and will therefore ( 42) be 
expressible in the form e G &, where G(0) is integral. 

Accordingly, the most general function of the required type is 



if there is a zero at the 



{(i - J) e^ + + --^}, 



50J THE GAMBIA FUNCTION 109 

Example. From Example 2, 48, we have 

1 -j-ao / 1 1 \ 

cot*-i=Z'( --- + -). 

_,. \z mr inrj 

Hence f(cots-i)<fe= 2? [' ( -^ + } dz - 9 

Jb\ z) _ Jo \z-mr mrj 

Iog !in = { log(l--l) + J-). 

2 -oo I \ ttTT/ ttirj 

Therefore sin = ff' f(l - )e^l = *fl(l - -. -A 

_,o vA mr/ ) i V **/ 

Note. "VVe cannot put nn**II'(l - z/mr) : for, since the series 2{l/(2 - WTT)} 

00 

is not convergent, the infinite product is not convergent. 

The Gamma Function. We define the Gamma Function F(0) 
by means of the equation 



where y is Euler's Constant. The expression on the right-hand 
side of the equation is integral, and has simple zeros at the 
points 0, ], 2, 3, ... : thus T(z) is holomorphic except at 
the isolated simple poles 0, 1, 2, 3, ... , and has no zeros in 
the finite part of the plane, (cf. 61). 

The Sigma Function. The method employed in the proof of 
Weierstrass's Theorem, when applied to Weierstrass's Zeta 
Function ( 48), leads to the integral function 



with simple zeros at the points 2??^ + 2 ?io> 2 , where in and n 
take all integral values. This is Weierstrass's Sigma Function, 
denoted by <r(z).' By logarithmic differentiation, it follows that 
<T '( Z )I ( ^( Z } = ^( Z )' As in the case of the zeta function, it can 
be shewn that <r( z)= o-(z), so that o-(z) is odd, (cf 76). 

EXAMPLES VI. 

1. Shew that the series 

1_1_J_4.1J_ 1 ! 
z 1! 2 + 12! 2 + 2 3! 8+3 

it presents a meromorphic function with poles at the points 
0, -1, -2, -3,.... 



110 FUNCTIONS OF A COMPLEX VARIABLE [CH. 

00 

2. Shew that the series ^z n jn\ represents a holomorphic function at all 
finite points of the plane, and deduce that 



3. If l.Kl, prove 



4. If z 1 lies within the circle of convergence of the series 2c n (z- 
shew that the Taylor's Series for the function at z l is Sc,/^-^)", where 



5. . Prove that the residues of e z at the origin and at infinity are both zero. 

6. Shew that, if A and B are the residues of e l/z z n /(l + z) at z = and z = oc : 

(i) 



7. Shew that the residue of e 2 log ^ at infinity is (e a - 

8. Shew that 



[Integrate e iz /(z 2 + z + l) 2 round the contour of Fig. 33.] 

9. If a >0, prove f dx * 

Jo I 

10. If > 1, prove ^ ^..-^^ 

11. If a > b > 0, shew that 

^-TsCaJaPIP). 



a + b cos 



14. If a and 6 are positive, shew thafe 

dx 



15. Shew that, if w^O, 



16. Shew that 



vi] EXAMPLES VI 111 

17. Shew that, if a and m are positive, 

P s * n2 .f l ' r ^' V 2 -.= . { e~ 2ma (2ma + 3) + 4wa - 3 }. 

18. If - 2r < c < 2?r, prove 

r 30 cosh ex ^ c 

Jo cosh 2 7r.r ' ~27rsin(c/2) 

19 Integrate gl g(l~ ?g ) round the contour of Fig. 33, and shew that 
(1 + 2z 2 ) 2 

I x ai ' 9 .,^= I -; ' ' =^^2 1). 

20. Integrate x ^ Log /, where 0<amp2<27r, round the contour of 
Fig. 38, and shew that 

/ y^ <L_c? l = 7T, / j-^- ' i ^-. 



" a sinh 2a 



23.' Prove 



2 1 20 22 2s 

and deduce - :=- 



*'" 

24. If ?i is a positive even integer, prove, by integrating ^ . round 
a suitable contour, that 



si H94 .97 dx e" l 

: ~. r == 7T" 



25. Prove cosecz = - + ( -1)"Y -+ V 

2-9i7T ?i7T/ 



26. Construct a function f(z) which is holomorphic except at the poles 
s= 1, 2, 3, ..., and is such that f(s) - z cot TTZ tends to zero at each of 
thesepoints. Am 1 Ig 



7T 

27. Prove 



J. Shew that t&uz = - 



(294-l)7rJ 






112 FUNCTIONS OF A COMPLEX VARIABLE [CH. vi 

29. Prove that, if \s\< 1, 

(!+*)(! 

30. Verify that : 



31. Prove that, if w n = 
ie product IIw /? is converg 

32. If \z \ < TT, shew that 



the product H iv n is convergent provided k = l and 2 






Deduce that, if < z < TT, 



CH. VII, 51] 



CHAPTER VII. 
VARIOUS SUMMATIONS AND EXPANSIONS. 

51. Expansions in Series by means of Residues. The theory 
of residues has been applied in 47 to the expansion of various 
functions in series of fractions. The following theorems enable 
us to shorten this process considerably. 

THEOREM 1. Let z = 'Re i0 lie on that arc of the circle js| = R 
for which O l = = 2 , and let zf(z\ as R tends to infinity, tend 
uniformly to the limit K, a constant, at all points of the arc, with 
the possible exception of the points for which ot e = = oc-f-e 
( e arbitrarily small). Also let | zf(z) = M, where M is finite, at 
all points of the arc. Then 



Lim 

For (Theorem I. 30), 
Limff f(z)dz + 

R_>^ U0j 

and 



a+e 



f(z) dz} = i(0 - 0J K - 2ieK ; 



Hence 



Therefore 



Lim 



T 'f(z)dz = 2eM. 

Ja-e 

*/(*)*].-<(4-4)K 

Lim f /(0)<7s = i(ft,-e i )K. 



'I'lu' tlieorem also holds if there is a finite number of excep- 
tional values of such as oc. 



Integrate f(z) = e iz /z round the contour of Fig. 5^, consisting 
of the positive x and y axes and quadrants of the circles | z | = 11 and | z \ = r. 
On tlie large circle, if e^# T --:>, :/'(:) !^e^ Ksllie ; so that zf(z) tends 

M.F. H 



114 FUNCTIONS OF A COMPLEX VARIABLE [CH. vn 



uniformly to the limit zero as K tends to infinity. Also, for all points on 
the quadrant = e --R*e< L . 



Therefore 
Hence 



/-/B 

Lim / f(z)dz = 0. 

R .tr, lan 



2 



so that, if the real and imaginary parts are equated, 
/"* cos x - e~ x 7 



Jo .r 2 




FIG. 52. 



Lemma. The function cot-Trs has simple poles at the points 



0, 1, 2, 



Let these poles be surrounded by circles of 



radius r, where r<[l/2; then a positive quantity M can be 
found such that cot TTZ = M for all points exterior to these circles. 
For (Examples III., 11), cot TTZ =\cot\nry : hence, if 

y = a, (a > 0), or = , | cot TTZ \ = coth ira. 
Now consider the region (Fig. 53) between the rectangle of 
sides x= rfcl/2, y= a, (a>r), and the circle s =r. In this 
region,- since | cot 7r2; | = cosh ^/^/(sinVaj + sinh 2 7r?/) 

cot 7T0 1 = cosh 7ra/sin(7rr/ x /2) if a? = r/^/2 or = r/ v /2, 
and |cot7r2;| = cosh7ra/sinh(7rr/ N /2) if y = r/j2 or =r/J2. 

Accordingly, since unity is a period of cot TTZ, \ cot TTS \ = M, 
where M is the greatest of the three quantities 

coth Tra, cosh 7ra/sin('jrr/ A /2), cosh Tra/sinh (7ny v /2 ). 
for all points of the -plane exterior to the given circles. 



51] 



LIMITING VALUES OF INTEGRALS 



115 



We leave the reader to prove that analogous properties hold 
for the functions : sec TTZ, cosec TTZ, tan TTZ ; 



S)TTZ 1 p"^Z I 1 


COST,? sinr2; COSTS; sin rz e rz + e~ rz 


cos TTZ sin7T0' sin7T0' COSTTS;' e nz e~" z 


where TT = r = TT. 




v> 








r 


^ 


J ' 


X' - 1 / 2 


^ 


} 






Y' 







FIG. 53. 

THEOREM 2. Let f(z) be a ineromorphic function, and let 
R x , R., , R ; , , . . . , be the radii of a series of circles with the origin 
as centre, no one of which passes through a pole of f(z\ and 

such that Lim R n = cc . Then if, as n tends to infinity, zf(z) 

ii ^>-'s> 
tends uniformly to the limit K for all points z = 'R n e ie such that 

6 l = = 0. 2 , with the possible exception of a finite number of 
sets of points . e = = a + , and if \zf(z)\=M. for all points 

on the arc. 



f0.> 
Lim f(z)dz = i(0 2 -e i )K. 



Tin- proof of this theorem is identical with that of Theorem 1, 
'xc-pt that Lim is replaced by Lim. 

R >ao >*> 

K.>'niiiplc.\. Let /(a)=cotir*/(f *) and K,, = ?i+l/2. Then, since 
Lim cot TTZ = - ?", Lim cot TT.S = ?', 

->x, y w 

it follows thai Lim. ?/'(:)=/ if ^^^TT- and Lim :/(-)=- '' if 



116 FUNCTIONS OF A COMPLEX VARIABLE [en. vn 

Also, by the Lemma above, \zf(z)\ = 'M. at all points of \z | = B H . Hence 

Lim 

and therefore, as in ^ 47, 



Example 2. If < r < 1 , prove 
& rx j * 2.^ 
x=+ ~~ 




52. Summation of Series by means of Residues. Since the 
idue of TT cot TTZ at each of its poles is unity, 



' 

where C is a contour enclosing the poles m, m-hl, ..., n, 01 

cot TTZ, and no others^/^j^is meromorphic, and 2 denotes the sum 
of the residues of TT o8E\ir$)f(z) at the poles of f(z) within C. 
Similarly 

y f(r)={ f^ dz v'=f ^ dz 2" 

where 2' and Z" are the residues of 
respectively at the poles of f(z) within" C. 

+ 00 1 'I 

Example 1. Prove 2 7 vt = "~2 * 

[Integrate TrcotTr^-. (x + z)~ 2 round the circle \z\ =B,, = M + l/2, and make n 
tend to infinity.] 

Example 2. If a is positive, prove 

V-M*-L e -7TH2/. 

- Va - 

Integrate e-^^Ke-^-l) round the rectangle of sides x- 
y= 1. Then, when m tends to infinity, 

+ =0 -+i 

2-*a 3 =- I 

,+c 



4 -- "S - 7r " 2/ ". (Examples IV. , 21 .) 



51,52] 



SUMMATION OF SERIES 



117 



Example 3. Gauss's Sum. Let S M = 2 T r , where T r =2irir/H. Then 

r=0 

T u _,.=T r ; so that S M =22 n , where 2 n stands for iTo + Ti + .-.+TV-i or 
^T + T 1 + ... + T M _o+|T H according as ? is odd or even. 

2 

Now, 2,,-r o or 2 n -|T -T (l/2 , as the case may be, is equal to the 
integral of **//(** -1) taken round the rectangle ABCD (Fig. 54) of 
sides #=0, #=w/2, y= E, indented at O and n/2. 



Yyl 








D 


C 




-- 

I 


- 

1 

f 




O 


I 

* 


1 

! 

*. 


/ 2 X 






But 
and 



A B 

FIG. 54. 

e 2nizynd z I |-A8 e -47rBx//' w 

CD C 2 8 _l' < J Q l_e-2irR^ C 47^R' 

2iriz2/n ^ T"/2 c 4irR.r/ /; 

.'ABe'- 7 "'--! " J ^"-"-1 " 4-l{ 



so that both of these integrals vanish when Tl tends to infinity. 

Again, when R tends to infinity, the sum of the integrals along the 
straight parts of DA tends to 



n'e-Zm^ln e-^iy-/" 
FSi^i + ^^l 



cos.r 2 o?^7-U sill .*-</.. 
Jo 

= ( 1 + 1) \/w/4. (^ 30, Example 5.) 
Similarly, the integrals along the straight parts of BC give 



/ " ' 



Kinally, the integrals along the small semi-circles at O and n/2 give -^T . 
and - .yr,i/2 or - .^T and 0, according as n is even or odd. Hence 



118 FUNCTIONS OF A COMPLEX VARIABLE [CH. vn 

Another Summation Formula is 



= ^ f 

-7T6 J 



where C is a contour enclosing the poles in, ra+1, . .., n, of 
cosec TTZ, and no others, and 2 denotes the sum of the residues 
of TT cosec (irz)f(z) at the poles of f(z) within C. 

Example. Shew that, if a is any non-zero real quantity, 



If a is small, the second series converges rapidly, while the first 
converges slowly. 

53. Roots of Equations. The following three theorems lead 
up to the proof of Lagrange's Expansion. - 

THEOREM I. If (j>(z) is meromorphic in a simply-connected 
region of boundary C, then, with the notation of 31, 



where A<1? denotes the total increment of amp {$(2)} when z 
describes C positively. 

17 ^< -^ 1 I (^0 ^ 1 A T 

r or 2*p 2*$ = ^: -. 1 , 'dz=- :ALo2Td)(0), 

27T^J C 0(2;) %7Tl 

where A Log <p(z) is the increment of Log 0(0) when z passes 
round C. Hence, if <j>(z) = 



But A log R = 0, since log R is uniform on C ; therefore 



THEOREM II. Let f(z) and 0(0) be holomorphic in a simply- 
connected region of boundary C, and let f(z) be non-zero on C. 
Then, if | <i>(z)+f(z)\<\ for all points on C, f(z) and f(s) + <t>(z) 
will have the same number of zeros within C. 

For, let w= l + <t>(z)jf(z)\ then, as z describes C, w describes a 
closed contour in the w-plane about it'=l, not enclosing the 
origin, and amp w returns to its original value. 

Hence the increment of amp {/(0) + 0(0)} is equal to the 
increment of amp {f(z)} ; and therefore, by Theorem L, these 
two functions have the same number of zeros within C. 



53,54] LAGRANGE'S EXPANSION 119 

THEOREM III. If f(z) is holomorphic for < r, and is not 
zero at the origin, a finite quantity p can be found such that, 
if w ^-p, the function \^(z,w) z wf(z\ regarded as a func- 
tion of z, has one and only one zero in the circle \z\ = r'< r : and 
this zero is itself a holomorphic function of w for w \ = p. 
For, let p be chosen so that, if \z\ = r, 

^(z, w}-^(z, 0) =>/(*) <r', 
provided ; i<; = p. Then 

(z,w)-+(z,0) ' 

+(*, 0). 

so that, by Theorem II., if w\^p, \f/(z, w) has one and only one 
zero, f say, within j z = r. 
Now, the integral 



_ f. 

-Trij 



. 

2-Trij \[r(z, w) 27ri Z wf(z) '" 

taken round j 5; = /, is a holomorphic function of it; ( 34). But, 
if w\^p, this integral has the value f ( 31, Corollary 2). 
Hence f is a holomorphic function of w for | ^ | = /o. 



COROLLARY. If F(z) is holomorphic for \z <^r, F(f) is a 
holomorphic function of w for | w = p, and 

F(f) 



wliere the integral is taken round | z \ = r, ( 31). 

54. Lagrange's Expansion. The results obtained in Theorem 
III. of the previous section can be stated thus : let f(z) and F(z) 
be holomorphic for \z\<r, and let f(z) be non-zero for = 0; 
then, if z denotes that branch of the function of w given 
by z = wf(z), which vanishes when w = 0, a finite region | w \ ^ p 
of the ^<;-plane can be found in which F(z) is hdlomorphic. The 
Taylor's Series for F(z) in this region can be found as follows. 

Let C denote the circle \z\-r\ then 



120 FUNCTIONS OF A COMPLEX VARIABLE [CH. vn 

since \wf(z)/z\<^l, 

if ? 

2iriJ 



since all the integrals in the first series have the value zero, 
(26, Cor. 8). 

In particular, if F(s) = 0, 



These are the well-known expansions of Lagrange. 
If the origin be changed to the point -f, and 0(z) be written 
for /(0 f ), these expansions become 



and 



where z is that root of z = g+w<f>(z) which has the value 
when w = 0. 



Example. Shew that the root of 2(1+2)"* = ^?, which is zero when v; = 0, 
is given by 



.. ,# i , . 

"2T 3! 41 

Rodrigues Formula for P M (f). If 2; is that root of 

*f+w("-i)A 

which has the value (=/=l) when u = 0, 



Before differentiating this series with regard to f, we must 
shew that a region can be found in the -plane in which the 
series is uniformly convergent. 



54] 



KODRIGUES' FORMULA 



121 



Let f be replaced in the expansion by \ = ik, where k 
is real and positive; then it follows from the theory of 
Lagrange's expansion that a value of p, say p-p l , can be 
found such that the series 



is absolutely convergent if \w 
strass's M Test, since 



2 



= p l . Accordingly, by Weier- 



provided | f \ = k, the series of equation (i) is absolutely and uni- 
formly convergent in w and f for | w \ = p l and for | f | = k ; and 
/c can always be chosen so that | f |<&. Hence (43, Th. 3) 



- 



Now 



where that branch of ,/(! 2w-}-w~) is taken which has the value 
1 when w = ; therefore 

1 + W " P - 46 > 



Hence, equating the coefficients of iu n in the two expansions 
for -^, we have Rodriguez' Formula, 



By differentiating the product (f 2 - l) n = (f - 1 ) M (f + l) w n times 
it can be shewn that the formula is also true in the exceptional 
cases f = 1. 

COROLLARY. 






S " 



122 FUNCTIONS OF A COMPLEX VARIABLE [CH. vn 

Example 1 . If m i= n, I* P IH (x) P n (x) dx = 0. 

.'-i 

For let m>n : then, by repeated partial integrations, 

(,)p,^ 



= 0. 

Example 2. Shew that P n ' 2 (^) <&,-= 2/(2w + 1). 

J- 

As in Example 1 we have 



-* 1 where A ' =2 * 1 - 



Example 3. Shew that 

s= A H P n () + A H _ a P II _() + A <l _ 4 P > ,_4(s)+ 
where A n =2 B (w!) 2 /(2w)!. 

Example 4. Shew that : 

(i) T 3 w P n (,s)cfe=0, where 

' 



55. Analytical Continuation. If f(z) is holomorphic in a 
region S, if ^(2;) is holomorphic in a region S', which includes S 
and if <l>(z)=f(z) for all points of S, (/>(z) is said to give the 
Analytical Continuation of f(z) in the region S 7 . 

00 

For example, the function f(z) = ^z n is holomorphic at all 

o 
points within the circle |z| = l, the function </>(z) = l/(l z) is 

holomorphic except at z = 1, and 0(0) =f(z) within z\ = l. Thus 
<j)(z) gives the continuation of.f(z) over the rest of the plane. 



Example. If f(z) = ^llz tl , over what region is/(0) holomorphic, and what 
function gives its analytical continuation ? ^4?is. Outside |0| = 1; !/( 1). 



54, 55] ANALYTICAL CONTINUATION 123 

The following theorems are useful in determining the ana- 
lytical continuations of functions. 

THEOREM I. If a holomorphic function f(s) and all its deriva- 
tives vanish at a point a, f(z) and all its derivatives will vanish 
at all points in the domain of a. 

For f(z) = ^c n (z-a) n , where c n =f n \a)/nl, (n = Q, 1, 2,...); 

o 

thus c = e 1 = c 2 =...=0, and therefore f(z) t f(z), f'(z),..., all 
vanish at all points of the domain. 

COROLLARY. If two functions and all their derivatives are 
equal at a point a, and if they are both holomorphic in a circle 
of centre a, they are equal at all points of the circle. 

For the differences of the two functions and of all their deriva- 
tives vanish at a. 

THEOREM II. If f(z) and all its derivatives vanish at a point 
of a connected region E in which f(z) is holomorphic, f(z) will 
vanish at all points of E. 

Let A (Fig. 55) be the given point, and P any other point of 
E. Let a path AP in E join A and P, and let d be the shortest 




FIG. 55. 



distance from any point on AP to the nearest singularity of 
A*) i s that the domain of any point on AP must be at leas 
of radius d. On AP take successive points A, P lt P 2 , P 3 , ..., 
such that each lies within the domain of the preceding point. 



124 FUNCTIONS OF A COMPLEX VARIABLE [OH. MI 

They can be selected so that, after a finite number of steps, a 
domain is reached which contains P. Then (Theorem I.) f(z) 
and all its derivatives vanish at P 1? P 2 , P 3 , ..., and therefore 
at P. 

COROLLARY. If two functions and all their derivatives are 
equal at a point of a connected region in which they are holo- 
morphic, they are equal at all points of the region. 

THEOREM III. If two functions f(z) and 0(0) are equal at all 
points of a line L in a region E in which they are both holo- 
morphic, the functions are equal at all points of E. 

For, if the points z l and z 2 lie on L, 



Lim = Lim 



Thus the first derivatives of f(z) and <j>(z) are equal at all 
points of L. Similarly all the other derivatives of f(z) and (j>(z) 
can be shewn to be equal at all points of L; and therefore the 
functions are equal at all points of E. 

This theorem is particularly important, as it enables us to 
extend theorems which have been proved for the real variable 
to complex values of the variable. For example, let 

and s = l: 



theu, if we assume that the equation 

sin' 2 # + cos 2 a; = 1, or f(x) = <f>(x), 

has been established for x real, it follows, since f(z) and 0(0) are 
holomorphic for all finite values of z, that /(z) = 0(z) for all 
finite values of z; i.e. that si 

Example 1. Prove P.(f)-A* 



Since the zeros of 1 2zf+{ 2 are i(l \/2), the expansion 



is valid if f!<\/2-l. Hence the- series of positive terms 2|P n (z)|R*, 
where R=0'4<\/2-], is convergent. 

But, if | z \ ^ 1, | P(*) 1 ^ | P M (t) |, ( 54, Corollary). 

Tims the series ^fr^-p^ 2P()f " 

is uniformly convergent with regard to both z and provided | z \ ^- 1, ! f | = H. 



55, 56] ABEL'S THEOREM 125 

Now differentiate with regard to z and in turn ; then 



so that ( 

Accordingly, if the coefficients of " are equated, 

P.fr)-,g-<L)-'* P r' ( '>. where |K1. 

Cfo CT2 

But the functions on both sides of this equation are holomorphic for all 
values of the variable ; therefore, for all values of z, 



Example 2. If ze bz = w, shew that 



provided ?0 | < e~ l /\ b \ . 

[Apply Lagrange's expansion for F(z)=e az . Since the series is convergent 
for 'w\<T l /\b\, it follows, by the principle of analytical continuation, that 
the equation holds for that region.] 

56. Abel's Theorem. A power series represents a continuous 
function at all points within its circle of convergence. If the 
series also converges at points on the circle of convergence, the 
following theorem shews that the region of continuity includes 
these points. 

XI 

THEOREM. If the power series ^c n p n = <f>(z) be convergent at 

o 
a point on its circle of convergence, and if z be a point within 

the circle, 

?e n z Q n = Um<j>(z), 
,. --^o ^ 

where z tends to Z along a radius.* 
Let ^ = /o(cosO-hisin 0): then 

c n z n = c tt p n (cos n + i sin nO) 



= 2 c />o naj " cos nO+i^Cnp^x" sin nO, 



where x = /o//o . < 

Hence it is only necessary to prove that if the series s^ 



wnere O = cco87 or 



is convergent, the function 2^a n x n will be continuous for O^x^l. 



* Cf. Brninwich, Infinite X-/Yx, 83. 



126 FUNCTIONS OF A COMPLEX VARIABLE [CH. vn 

Now let r llt p = a H+l + a n+2 + . . . -f a n+lt . 

Then for any particular value of n, finite quantities H and h 
can be found such that H>r Mf ,,>/*, where p is any positive 
integer. Hence 



a. 



tl p - r n p 



-a^ 



if 0a; 



Similarly, a, l+1 ^ i+1 + a H+2 fc+ 2 + . . . + a B+Jl aj N +J' > //.. 

But an m can always be found such that | H |<e and | h 
forn^m; so that l^^^a^^^/.^o,^^ < 

Hence the series converges uniformly for O^a-^1. Conse- 
quently the theorem is proved. 

COROLLARY. If the series converges at all points of an arc 
of the circle of convergence, <j>(z) will be continuous on that arc. 

Example 1. Consider the function 



rendered uniform by a crosscut along the negative real axis from - 1 to - oc . 

V, 




FIG. f>(j. 



Let A and P be the points - 1 and z respectively, and let the circle of 
centre A and radius AP cut OX in E (Fig. 56). Let the integral be taken 
along the path ORP : then 



where > denotes z_OAP. 



56] SUMMATION OF TRIGONOMETRICAL SERIES 127 



Now, if |*1<1, 

and this series converges at all points of the circle |.*| = 1 except A, (Abel's 
Test, 38). Thus it represents the continuous function log(l+^) in the 
region of Fig. 56, bounded by the circle \z\ = l indented at A. Accord- 
ingly, if P is the point z = e ie , where -TT< #<TT, 



so that, if real and imaginary parts be equated, 
cos 20 . cos 30 
3 



^ UUS^f UU t>l/ 

COS r- "I -... = . 



rO] 

2 J 



2 

. sin 20 sin 3 
ui6 2" + ~3~ 

^ = in the first equation gives log 2= 1 - 1/2 + 1/3- .... 
The two series 

of 38 are uniformly convergent in the interval e^0 = 27r e, 



They can therefore be integrated term by term. 



H '-ample 2. If - TT g^ ^ zr, prove 

. COS26' COS 36* 



0- 



3. If A, B, P (Fig. 57) are the points ?', i, and z respectively 
shew that, for all points of the region bounded by the circle \z\ = l indented 
at A and B, 




128 FUNCTIONS OF A COMPLEX VARIABLE [CH. vn 

Deduce 

, cos SO cos 50 f ^M, if cos is positive; 

(i)cos0- + _ l _...= , o, ifcos<9=0; 

[ - 7T/4, if cos is negative : 



EXAMPLES VII. 

1. Integrate -{- -e~ 2 J round the contour of Fig. 52, and shew that 

' 



2. Prove 

3. P,ove 



4. Prove ] -UU* ^ 



5. If - TT ^ r ^ TT, shew that : 

1 
2 



(i) ^T e ''" + 6 ~'' 2 _ 1 1 cos r cos 2r cos 3;- 

' 8 - -" z ~ 2 2 



cosr cos2r cos3/ % 



6. If - - < r < TT, shew that : 



(i) ? g r2 e~" sin r 2 sin 2r 3 sin 3r 



/\ TT sinrz sinr 2sin2r 3sin3r 

* "V-1+ 38-4 a-9 



7. Shew that P/ 

Jo .r 2 

8. If - 6 < ' < />, shew that 

/.% p /"" sin aA' dx TT sinh 

,'o sin bx 1 +,v 2 2 sinh b 

cos ax x d% TT cosh a 



("\ P / 

Jo 



(iv)P 



, /*" sin era? dx _ TT sinh a . 
.'o cos bx x(\ +,v 2 ) 2 cosh b ' 
cos ax dx TT cosh a 



(v)P 



.'o cos bxl+x* 2 cosh 6 ' 
""sin ca? .rc?^; TT sinh 



Jr> cos fejt; 1 +# 2 2 cosh // 



56j EXAMPLES VII 129 

9. If m > 0, prove 

f* 3 dx 

I cos 4mx tanh x = log (coth ?/ITT). 

10. If a and b are real, and TT < b < TT, prove 

r cosh fca: cosh(/2)cos(6/2) 

/ r - cos ax dx = -- \^- - ^4-** 
Jo cosh TTX cosh a + cos 6 

11. Prove 



12. Shew that, if m is a positive integer, the root of z= + wz m+l which has 
the value when w=0 is given by 

+ 3) . . . 



| ^2m + l , ^ , . . . ^ nm + l | 

provided | w \ < m m (m + 1)-"*- 1 1 ^|-' rt . 

13. Prove that the coefficient of z tl ~ l in the expansion of {2/(e*-l)} H is 
( - I)"- 1 . [Use Lagrange's Expansion for w=e z - 1.] 



14. Prove that the coefficient of z n ~^ in the expansion of 
is 1/2. [Use Lagrange's Expansion for iv=z(2 + z)/(l + z) 2 , F(2)=log(l +z).] 



15. If m and w are distinct positive integers, shew that 



1, Prove 

17. Prove /P-+i(*)P^(^^ 

18. Prove 



19. Shew that n_^^ = (2 . ? _ 1)Pn _ l(4 

20. If 1 10 <l/4, shew that 



where w=z(l z). 

21. If [ '"|</3- 1 , shew that 



where z is that root of log z = wz which has the value +1 when w= 
[In Ex. 2, 55, puta=l, b= -l,^= 

22. If n is a positive integer, shew that 



[In Ex. 2, 55, multiply by e$~~, and equate the coefficients of 2".] 

M.F. I 



130 FUNCTIONS OF A COMPLEX VARIABLE 

23. If is real and | 1 < e~\ shew that : 

1 2- *3 3 

(i) cos 0=1 -0sin0 + ^0 2 cos 20 + ^03 8^30-^04 cos 40-.. 



[CH. 



[In Ex. 2, 55, put a=b=i.\ 

24. Prove that, for all points within and on the circle | z \ 1, 



and deduce that 

... cos 20 cos 30 cos 40 



w 1.2 2.3 ^ 3.4 

/ /9\ fi 

= (l+cos0)logf 2 cos-J -cos 0-^sin ; 

.... sin 20 sin 30 sin 40 



2 



/ n\ a 

= sin log (2 cos - } - sin + -5 (1 + cos 0). 

r 

Jn 



=log2. 



25. Prove 

[Integrate 
561 ' 

26. If - 1/2 < m < 1/2, shew that 

f 30 sinh 2 wi# , , , 

^r dx = i log sec mir. 
Jo x sinn x 

27. Shew that, if < amp z < 27r, 



Deduce : 



round the contour of Fig. 33, and use Ex. 1 



Graph the functions represented by these two series for all values of 0. 
28. Shew that 



sin 30 sin 50 
H -- ~ 



29. Shew that 






7T/4, if sin 0>0. 
_ . . /, 
^ sin 0=0. 

-7r/4, if sin0<0. 

!T /9 if _! r -< #<iT 
4V 2=^ = 2' 



vii] EXAMPLES VII 131 

30. Shew that, if - Tr/3 < < Tr/3, 

f. COS 50 COS 16 COS 11 # _ 7r v /o~ 

^5~~ ~T~ 11 = 6 

31. Shew that the locus represented by 

I tlt'sin^sin^O 

consists of two orthogonal systems of straight lines dividing the (#, y) plane 
into squares of area 7r 2 . 

32. Shew that the equation 

2 I sin ttycosft#=0 

represents the lines y=mir, (ra = 0, 1, 2, ...), together with a series of 
arcs of ellipses whose axes are of lengths TT and 7r/\/3, placed in squares of 
area 7r 2 . Draw a diagram of the locus. 

*> / _ ly*- 1 

33. If f(x, y, z) = 2 i 1 sin nx sin wy sin 7?^, 

i=l W 

shew that, within the octahedron bounded by the planes xyz=Tr> 

34. If 



shew that, for < 6 < Tr/6, r=2 cos (0 + 7T/3). 

Graph the curve foi 1 values of between and 27r. 

35. If w = z(l +2 2 ), and the principal value of tan" 1 * is taken, prove that 

4 w 3 6 . 7 w 5 8 . 9 . 10 w 7 
ten-if-^-jy 3-+-^ T - ^y 7 +-.. , 

provided |w|<f\/3. 

36. Shew that the two functions 4? tan 2 each possess only one zero 
within the unit circle. 



[CH. vni 



CHAPTEE VIII. 
GAMMA FUNCTIONS. 

57. The Bernoulli Numbers. The Bernoulli Numbers B 1? 
B 2 , B 3 , ... , are defined by the expansion 

v * T^ 

= i - + VC IV 1 - 1 - n ^z Ln 

e*-} 2 + 2/ (2w)| * 

considered in 44. Their numerical values can be found by the 
method established there ; thus 

"i = ib ^2 = inr BS = TV> 

From the expansion it follows that ( l) n ~ 1 B n /(2ii)! is the 
residue of l/{z* n (e*-l)} at = 0. But (Theorem II, 51) 



where c v denotes the circle \z = (2i/+l)7r. Therefore 

T5 oo -I co 9 

(_l)n_E!!_ = y/ L, = (_l)ny ^_ 



".-' j- 

Example. Shew that J^+i^ + J^ + ... = |:. 

T/te Bernoulli Numbers as Definite Integrals. If a is positive, 
.+6-'' a = ?r cot OTS). <?-"<&, (52) 



'C 

where C denotes the rectangle ABCD (Fig. 58) of sides x = 0, 
x=b=n+ 1/2, y = R, indented at O. 

The integral of (l/2i)cot (TTZ). e~ a " along the small semi-circle 
tends to -J, (30, Th. 2). On the contours ECDF and GABE 

replace (l/2i) cot (TTZ) by 

i i 

and A-f n - 



57] 



THE BERNOULLI NUMBERS 



133 



respectively. The integrals arising from the terms J and + J 
tend to p P 

^ e~ az dz and ^1 e~ az dz 

"JECDO "JOABE 

respectively. But, since e~ az is holomorphic in the rectangle, 
each of these integrals is equal to 



Thus we find 



where 
and 

Now 
so that 



p 

= J 






- 



ax dx. 



Y, 



' 1 



i t 



A B 

FIG. 58. 






Hence 



LimI 1 = 0. 



134 FUNCTIONS OF A COMPLEX VARIABLE [OH. vm 

2 f 6 2 

Again, 1 1 5 ' 



LimI = 0. 



.-a = l_l^r2sinat/ 
a 2 



Hence 

Accordingly 

But 



Hence, expanding sin at/ in powers of a (cf. Bromwich, Inf. 
Ser., 176, B), and equating coefficients, we have 



B, 






Example. Prove 

58. The Asymptotic Expansion of Euler's Constant. Let 

O Til I L. \AJ% / Q C? O \ 

where C is the rectangle ABCD (Fig. 59) of sides o;=l, x = 
y = R, with small semi-circles at 1 andvjt,. 

D C 



i t 



FIG. 59. 



The integrals of (l/ / 2i)cot(7r2;).2;- 1 along the small semi-circles 
at z= 1 and z = n tend to J and l/(2n) respectively (30, Th. 2). 
On the remaining portions of C replace (I/2i)cot(7rz) by 

' 



e -zmz_ 



or 



58] EULER'S CONSTANT ASYMPTOTIC EXPANSION 135 

according as z lies above or below the #-axis. The integrals 
arising from the terms ^ and + i each tend to 

" 



Thus we find 



= l \ l \ ( ndx i r 2y d y f R 

2~ h 2tt~ h J 1 aj+Jol+^e 2 "*-! J 



T _ dx C n 1 dx 

- 



v 2 dx 

-- 



so that Lim 1 = 0. 

R-oo 

Hence 



S . 

But f _?SL _^_/l rgy^y - 

Jo 7i 2 + y 2 e^-l^?i 2 Jo e 2 ^-! ' 



therefore, as n tends to infinity, S n log n tends to the limit 
If" 2y q 



This limit is Eider's Constant, and is denoted by y. Thus 



Bo 



where 



= r 2y2k+i d y ^ i r 

Jo W 2 *(w 2 + y 2 ) e 2 ^- 1 < ^'+^ J 



136 FUNCTIONS OF A COMPLEX VARIABLE [OH. vm 

00 

V* -I / r 2fc+2 

Now B w = (2fe+l)(8fc+8) r 4r 

" 

r=l 



but lr*Slt*=^6 and lr 2 *+ 2 >l, so that 



CO T> 

Thus the infinite series 2 ( I)*' 1 srA. is divergent 
^i 2kn 2k 

Nevertheless, if sufficiently large values of n are taken, the 
sum of a few (say k) terms will give the value of y to any 
approximation required. For R^ can be made arbitrarily small 
by increasing n. An expansion such as this, consisting of 
a finite number of terms and a remainder which can be made 
arbitrarily small by sufficiently increasing the variable, is called 
an Asymptotic Expansion. 

Example. If n = 10 and k = 2, shew that E fc < '000000004. 

59. Convergent Integrals. In our definition of an integral 
we assumed that the path of integration did not pass through 
a singularity of the integrand /(). It is sometimes possible, 
however, to extend the definition to include cases in which 
an extremity z l of the path is a singularity of f(z). 

Let %' be a point on the path of integration ; then, if the 

ft 
f(z)dz tends to a definite value as tends to z lt this 
z (X 

limit is taken as the value of I f(z)dz, and the integral is said 

JZo 

to be convergent. The necessary and sufficient condition for 

Jz" 
f(z)dz should tend to zero as z and z" tend to z r 

The following two rules are useful for determining the con- 
vergency of integrals. 

RULE I. Let z l be a finite point ; then if a number n < 1 can 
be found such that (z z^) n f(z) tends to a definite limit L as z 



tends to z l , the integral I f(z) dz will be convergent. 



59, 60] CONVERGENT INTEGRALS 137 

For if z be chosen so that | (z z^) n f(z) L < e, provided 
|0 Zi\ = k, where k= z' z^\ t 

*! d/) , where P =|*-,| 



and this quantity can be made arbitrarily small by decreasing k. 

Example. Shew that the integral f~ 2 -r is convergent. 

-fci vUz-ZiX 2 -^)} 

RULE II. Let the point z l be at infinity ; then if a number 
n > 1 can be found such that z n f(z) tends to a definite limit L 
as z tends to infinity along the path of integration, the integral 
will be convergent. 

For if z' be chosen so that, for points z on the path of integra- 
tion between z' and infinity, z n f(z) L | < e, 

, where >=s, K= 



|L|+e 



and this quantity can be made arbitrarily small by increasing K. 

Example. Shew that the integral / e~ z z n dz, taken along a straight line 
making an angle < with the .r-axis, converges if ?r/2 < (f> <7r/2, and n> 1. 

60. Uniformly Convergent Integrals. Consider the integral 

ff(z t f ) cZ0, where /(0, f ) is holomorphic with regard to both 3 
c 

and f at all points of a region A in the z-plane which contains 
the curve C and at all points of a region A' in the f-plane, 
except for a singularity at the (upper) extremity % of C.* 
Let z be a point on C, and let C t be that part of C which 
has z' as its (upper) extremity. Then if, for all points f of A', 

I f(s, Qdz tends uniformly to the limit 0(f) as z tends to z lt 
the integi'al is said to be uniformly convergent in A r . 

* It is assumed that the path C is independent of f. 



138 FUNCTIONS OF A COMPLEX VARIABLE [OH. vm 

THEOREM. If f(z, )dz is uniformly convergent in A', it is a 
J c 

holomorphic function of f at all interior points of A', and 

ajjsf A* f)<fc=J |^/<*. f)<fe, <=i, 2, 3, ...). 

Let z' be chosen so that, for all points f in A', | >/ 1 < e, where 



Then 0(f) is continuous in A', since \Kf) is continuous (34) 
and |i;|<e. 

Again, let f be any interior point of A 7 , and let K be the 
boundary of a simply-connected portion of A' of which f ' is an 
interior point. Then 



o that 






.+ 
- eL 



where d is the shortest distance from f ' to K, and L is the length 
of K. 

Hence I =~= f(z t ?)dz or \!s M (n 



converges to the limit ^ -* I rffT^i as s' tends to 2 l ; and 
therefore 



In particular, if ?i = 0, /(^, f')^ converges to the limit 

Jo 



, S o that 0(r) = - - Now this integral 

^TT'UK ^ 

is holomorphic (35, Corollary 2) at f. Hence 0(f) is holo- 
morphic at f ', and has the derivatives 



Example 1. Integration under the Integral Sign. 

If C' is any path in A', f <(0^= f ( /(*, 
Jc'* ~'c J<y 



60] DIFFERENTIATION UNDER INTEGRAL SIGN 139 

For ( <t>(Qd=l VKCK+f rjdf 

Jc' Jcr JGf 

= 11 fa K<fe+ f V<%, (34, example). 

JCi J0 Jc' 



Now / 7?d <eL, where L is the length of C'. Hence / / /(z, {)ddz 

\ Jc' J<h JO 

tends to the limit / </>(O^C ' so that 

-'C' 

f <KK= f f /(*, Mi**. 

Jc 1 Jc Jc' 

The following two rules, the proofs of which are similar to 
those of 59, are useful for determining the uniform convergency 
of integrals. 

RULE I. Let the extremity z l be a finite point ; then if, for 
all values of in A', a number n < 1 can be found such that 
(0 z^ n f(z, f) tends uniformly to the limit L(f) as z tends to z lt 

the integral I f(z, )dz will be uniformly convergent. 
Jc 

RULE II. Let the extremity considered be at infinity; then 
if, for all values of f in A', a number n^> I can be found such 
that z n f(z, f) tends uniformly to the limit L(f) as z tends to 

infinity along C, the integral I f(z, g)dz will be uniformly 
convergent. 

Example 2. Consider the integral 4>(z)= / e^^dt^ where R(2)>0. 

Jo 

Let a ^ x= R(2) ^ 6, where a > ; then, if * > 1, 

| re-*?' 1 1 = e-$* +n - 1 ^ e- f t b+n - 1 . 
But Lime~ i! ; 6+n - 1 = ; hence the integral converges uniformly at its 

t >-QO 

upper limit. 

Again, if t < 1, | t n e-*t z - 1 \ ^ e~'r +n - 1 . 

Now choose a and ?* so that a < I and (1 - a) < < 1. Then 



hence the integral converges uniformly at its lower limit. 

Now, if R( 2 )>0, a and b can be chosen so that a^~R(z)^b ; hence 

isholomorphicfor R( 2 )>0 and has the derivative f e^^ 

Jo 

It is easy to verify, by partial integration, that: (i) 
(ii) </>(!)=! ; (iii) if m is a positive integer, <j>(m 



140 FUNCTIONS OF A COMPLEX VARIABLE [CH. vm 

Example 3. Consider the integral 



taken along a straight line which makes an angle ^ with the .r-axis. If 
satisfies the inequalities 



where r is positive, all the values of f are excluded which make 2 +2 2 =0. 
Now, if \z\=p>Il, 



fel'g* <"^ 



which tends to zero as p tends to infinity if l<w<2 ; hence the integral 
converges uniformly at its upper limit. 
Again, if \z\ = p<r, 



which tends to zero as p tends to zero if 1 > n > ; hence the integral 
converges uniformly at its lower limit. 

Accordingly, <(f) is holomorphic, provided ^r ir/2 <ampf<^r + 7r/2, 



Example 4. Consider the integral 



taken along the path of Example 2, where - 7r 
Let f be confined to the region defined by 



Then, if || = / 



ir/2. 



where x ( = /acos^) tends to infinity with />. 
Now, if n< 2, 



hence the integral converges uniformly at its upper limit. 
Again, log ( 1 _^ 27r? ) = - log z + log/(s), 

where /(0) is holomorphic at 2=0. Therefore, by Example 2, the integral 
converges uniformly at its lower limit. 

Accordingly, <() is holomorphic, provided 



60, 61] THE GAMMA FUNCTION 141 

61. The Gamma Function. Gauss's Definition. Let T(z) 
denote the function 

n ! n z 



Then 



so that this definition is equivalent to that of 50. 

The following properties can easily be deduced from the 
latter definition : 



(ii) r(m + l) = m!, where in is a positive integer; 
(iii) T(z)T(l-z) = -^; 

SmiTZ 

(iv) the .residue at z = in, where in is zero or a positive 
integer, is ( l) m /m!. 

The Function ^r(z). Similarly, if \[s(z) = -=- log T(z+l ), we have: 

dz 



(ii) ,/,((>)= -y; 
(iii) ^() = l+l + ...+l 

(iv) \fs( z l) = \ls(z) + 7r 

Gauss's Function II (z). The notation 11(2;) is frequently used 
instead of T(z + 1) : thus 

U(z) = zli(z-l\ n(m) = m!, and U(z-l)TL(-z) = Tr/sm TTZ. 
Euler's Definition. The Gamma Function may also be defined 

as the integral I e'^ 3 ' 1 ^, provided R(2;)>-0. We shall now 
Jo 

prove that the two definitions agree for values of z which satisfy 
this condition. 



142 FUNCTIONS OF A COMPLEX VARIABLE [CH. vm 

If R(z)>0, and n is a positive integer, then, by partial 
integration, 



+ 



Thus, writing y u/n, we have 



so that r(2;) 

Now let /(t*) = 1 - & (l - -Y, 

\ lu/ 

where Q^-u^n', then 



Thus /(^) is an increasing function ; so that 



I- >Q, or 6-^(1--) . 
n/ ~ 



Again, 

fu fw / ^,\n-l^, x,w fu 

f(v)dv=\ ^(l--) -^<- 
o Jo \ n/ n nJo 

Accordingly, if ^ u ^ n, 

(fii\n n,2 
1-) <1L. 
W ~2n 
Now we can write 



vdv e u -^~ 



o 

f'' f'' / ii\n 

- e-"^- 1 ^- [l-.-W- 1 ^ 
o \ Ja J\ n/ 

Let a be chosen so large that, for all values of n greater than a, 

j 



n 

e- u u z - 

a 
Cn / n, \ ?i 

and therefore ( 1 -- ) u z ~ 1 du 

J \ n/ 



< e ; 



61] 
then 



EULER'S DEFINITION 



143 



f/t 
o 



n 



Hence, if R(V)>0, 

r(z) = Limf'Yl--Y l 
-*J<A n/ 



Gamma Function expressed as a Contour Integral. 
Euler's expression for F(0) can be replaced by the following, 
which holds for all values of z. 



Consider the 



integral I e~ 
Jc 



t where C is the contour of 



Fig. 60, with its initial and final points at infinity on the posi 




FIG. 60. 



FIG. 61. 



tive -axis. The initial and final values of amp f are taken to 
be and 2?r respectively. 

Now replace C by the contour of Fig. 61, consisting of the 
-axis from + oc to e, the circle | f | = e, and the -axis from e 
to +x . Then, if R(V)>0, we have, when e tends to zero, 



Now the functions on both sides of this equation are holo- 
morphic for all values of z. Hence the equation holds for 
all values of 0, and 



1. Prove that 



_ 
1 () 



where C denotes ;i ]);itli which starts from - oo on the J-axis, passes round 
the origin in the positive direction, and ends at - oo on the -axis. The 
initial and final values of amp are taken to be -TT and TT respectively. 



144 FUNCTIONS OF A COMPLEX VARIABLE [CH. vm 

Example 2. Gauss's Theorem. IfR(y-.-/3)>0 



For ( 36, Example 2) 
F(oc, ft y, 1) 



xF(a, ft y + rc+1, 1). 
C 



= Lim 
-~ 
and Lim F(o., ft y + w + 1 , 1) = 1. 



Example 3. If E (y ) > 0, R (y - a. - /?) > 0, shew that 

F(--, -ft y-a.-ft l) = F(oc,fty, 1). 

62. The Beta Function. Consider the integral of 

f(z) = zP-\l-z)<i-i 

taken round a closed contour which starts from a point A 
(Fig. 62) on the #-axis between and 1, and is composed of : 




FIG. 62. 



(i) a circuit APA round z = 1 in the positive direction ; 

(ii) a circuit AQ A round z = in the positive direction ; 

(iii) a circuit ARA round z = 1 in the negative direction ; 

(iv) a circuit ASA round z = in the negative direction. 

After describing this contour /( z) returns to A with its initial 
amplitude, which we assume to be zero. The integrand is a 
multiform function ; but since, at every point of the path, the 
branch integrated is uniform and continuous, the definition of 
26 holds for the integral. The notation 



rd+.o+.i-.o-) 
f(z)dz 



is used to denote this integral. 



61, 62] THE BETA FUNCTION 145 

The path APA can be deformed into the contour consisting 
of : the ic-axis from A to 1 e, the small circle z 1 1 = e described 
positively, and the #-axis from 1 e to A. Such a contour is 
called a (positive) Loop. If it had been described in the opposite 
direction, the loop would have been negative. Similarly the 
circuit AQA can be replaced by a positive loop about the origin, 
and the circuits AR A and ASA by negative loops about z 1 and 
z = () respectively. As z describes the circular part of the first 
loop, the value of f(z) changes from f(x) to f(x)e 2qiri ; similarly, 
the descriptions of the circular parts of the other three loops 
give/(z) the values J-(x)e 2( ^ + ^ iri ) f(x) e* piri , and f(x) respectively. 

We now make the radii of the circular parts of the loops tend 
to zero ; then, if p and q are real and positive, 

f(l+, 0+.1-.0-) 

*-i(l-.*)r-i<Zs 

J p 

* 1 ***} xP- l (l -x)*- 1 dx 

Jo 
, q) 



Now the functions on both sides of this equation are holo- 
morphic in p and q ; hence the relation holds for all values of p 
and q. Accordingly, if we define ~B(p, q) by the equation 



we have, for all values of p and q, 

m+,o+, i-,o-) 

Example 1. With the same initial conditions, shew that 

/(!+, 0-.1-.0+) 



Example 2. By means of the transformation #=(2-l) 2 , shew that, 
if R(p)>0, 



Deduce that, for all values of p, 

(i) 



The latter equation gives the Duplication Foi-mula for the Gamma 
Function. 



I.F. 



146 



FUNCTIONS OF A COMPLEX VARIABLE [CH. vm 



63. The Asymptotic Expansion of the Gamma Function. 
From the expression 



we derive the equation 



Now let C be a closed contour (Fig. 63) consisting of a semi- 




FIG. 63. 



circle of radius p + 1/2, where p is an integer, part of the ^/-axis, 
and a small semi-circle at O ; then, if R(2;)>0, 



V 1 J_ 

V(^ + ^) 2 ~2 7 ri 



The integral of (l/2i) cot (TT^ ) . (z+)~ 2 round the small semi- 
circle tends to l/(2z 2 ), ( 30, Th. 2). On the remaining part of C 
replace (l/2i) cot (xf ) by 



or 



according as I(f)<0. The integrals arising from the terms I 
and -f \ each tend to 






63] ASYMPTOTIC EXPANSION OF GAMMA FUNCTION 147 

Thus we find 

V l 1 

W 



where Lim 1 = 0. Thus 



Hence 



(60, Ex. 1), 

where the constant K must be real, since all the other terms are 
real when z is real. Therefore 



where K x is a real constant, and 



(60, Ex. 4) 
Now, since T(x 



so that K=-(a5-f-i)logl + - + l+ J(aj)-J(aj 



hence LimJ(oj) = 0, and therefore K = 0. 

X >-co 

Again, since T(z)F(lz) = Tr/simrz, 

TQ I iiQiy iu)= ^ e ~ 
Therefore 



where tan~ 1 (2it) denotes the acute angle whose tangent is 2u. 
- f log(l-e-2m ? )^ 77= _^_r i iog(i_ a; )^ j where x = e-27rr, 



118 FUNCTIONS OF A COMPLEX VARIABLE [OH. vm 

Now, if x and y are positive, 



xdrj 



-vX^ + rf- 

Thus LimRJ(J + m) = 0; so that (40, Ex. 2), K' = logv/27r. 



Therefore 

Again, let J^(z) denote the integral ( 60, Example 4), 



taken along a straight line making an angle \]s with the -axis, 
where 7r/2 < i/r < 7r/2 and z=f=Q. Then, since, for values of 
amp f between and \/r, 



tends uniformly to zero as f tends to infinity, J^(a?) = J(cc 
(30, Th. 1). 

Now J^(0) is holomorphic for the region R^ defined by 



Also, corresponding to any point z for which TT < amp z < ?r 

YA 




FIG. 64. 



and z=f=Q, & value of i/r can be found (Fig. 64) such that the 
positive aj-axis and the point z both lie in R^,. Accordingly, by 



63] 



CONSIDERATION OF THE REMAINDER 



149 



the principle of analytical continuation, since T(z) and Iog0 
are holomorphic provided ?r < amp z < TT and z=f=Q t 



= log \/27T + (Z %) log 



where J. W - 

But,(30,Th. 1), 



Hence 



Also the least value of |dfif | is the perpendicular distance 
of z from the line uQvJ (Fig. 64). Thus, zi\^\\z\, where 
X = cos ((j) i/r), (amp z = <p)', so that < X = 1 ; hence 



1 



Therefore 



1 



j^-L^ldfl, (39,Ex.l) 



B 



n+ , 



The infinite series 



B 1 B 1 



is divergent (cf. 58) ; but J n (z) can be made arbitrarily small 
by increasing z, so that, for sufficiently large values of z\, 



150 FUNCTIONS OF A COMPLEX VARIABLE [OH. vni 

a finite number of terms gives the value of the function to 
any approximation required, provided ?r < amp z < TT. The 
series is therefore asymptotic. 

COROLLARY 1. For all values of amp z such that 



- 1/2 e- z ) tends uniformly to the limit unity as z tends 
to infinity. 

COROLLARY 2. If z is real and positive, we can take i/r = 0. 

Then X = l, and T> T 

I T (z\ I ^ " 

*W I ^ 



so that the remainder after any term in the series for logT(z) 
is numerically less than the succeeding term. 



COROLLARY 3. m! = \/27rra( j e 12m , where < 6 < 1 : this is 

_ /771\ m 

known as Stirling's Formula. The expression \f2jnni J is 
usually spoken of as the approximation to m! when m is large. 



Example I. Prove tan- 1 1 _ ^ =\(\ - 

where the principal value of tan" 1 ^ is taken. 
Let C denote the rectangle of Fig. 59 ; then 

log(l .2.3...w)=T- . 

ATTl c 

Hence, by the process employed in 58, we obtain 



Again, log(l . 2.3 ... 




so that 



Hence, if ?i tends to infinity, we have 

fStM 

Example 2. She w^ that, if - ?r < amp z<ir, 

,-a =1. 



63] THE HYPERGEOMETRIC FUNCTION 151 

Example 3. Shew that, if - ?r/2 < amp < Tr/2 and f =f 0, 

j a+xi ^ r a+Xi {* TTdz_ _ 

2irtV_ fl _., U ^ 2iriJ- a -i r(z+l)iin~~' ' 

where a> and the path of integration is a straight line. 
If z=~ReM, where -7r<0<7r, 



Lim 

2->00 



! 1 

= Lim i -r,. . = Lim 



1 

IZ^zW* 

:Lim . g R cos 0(1- log R)+R sin 0.0 



Hence, if = pe^ and 6 + 0, 



=v / 2^Lim A- gBcos(l+logp-logR)+Rsin(-*Tir) 
^ 



Lim 

,_>*> 

according as sin is positive or negative. 

Accordingly, if 7T/2 ^ 6 ^ e, or if - Tr/2 ^ (9 ^ - e, or if - a ^ R cos ^ ^ 0, 



tends uniformly to zero as z tends to infinity. Thus (60, Rule II.) the 
given integral is uniformly convergent. 

Next, if -e^ifl^e, let z=H m e i0 , where R m =m + l/2 and ra is an integer ; 
then 

Lim 



- 

+ 1 ) Sin 7T2 

where 2M ^ ! cosec 772; ! ( 51, Lemma). Hence 



tends uniformly to zero as m tends to infinity. 

It follows ( 30, Th. I.) that the given contour can be replaced by a closed 
contour consisting of the line x -a and that part of the circle |s| = 
where m may be increased indefinitely, which lies to the right of this line. 

Now the only poles within this contour are those of 1 /sin TTZ ; hence 



TT^Z _ _ i * _ 

~ + " 



Example 4.* Shew that the integral 
2^'J 



where -TT< amp(- ^)< TT, and the integral is taken upwards along a 
straight line (Fig. 65) parallel to the y-axis, with loops, if necessary, to 
ensure that the poles 0, 1, 2, 3, ... , are to the right of the contour, while the 
poles -a., -0.-1, -0.-2, ..., -/?, -/?-!, -/3-2, ..., are to the left of 
the contour, is uniformly convergent. Negative integral and zero values 

*Cf. E. W. Barnes, Proc. Land. Math. Soc. t Ser. 2, Vol. 6, Parts 2 and 3, 



152 FUNCTIONS OF A COMPLEX VARIABLE [CH. vm 



of <x and ft are excluded, since the curve could not, under such conditions, be 
drawn. Also shew that, if | f < 1, the integral has the value 



while, if | f | > 1, it is equal to 



-a * -a-3 -a-2 



-jS-2 -/3-1 -j8 



X 



- yj 



Y 

FIG. 65. 



Firstly, let - =/>e^, where p< 1 ; also, let z=~Re iQ . Then, if e ^ O^TT/^ 
or if - 7T/2 ^ ^ ^ - e, where tan e < i log (1/p), 



Lim 

according as sin 6 is positive or negative. Accordingly, since TT 
T( -*)<-) or -^ 



r(y+z) 

tends uniformly to zero as z tends to infinity. Thus the given integral is 
uniformly convergent. 
Again, if - e ^ 6 ^ e, 



so that, if R = m-f 1/2, where m is integral, 



63] ANALYTICAL CONTINUATION 153 

tends uniformly to zero as in tends to infinity. Hence it follows, as in 
Example 3, that the integral has the value 



oL o 
r(y) 



Secondly, let p > 1 . Then, since 



T(l~y-z)T(-z) 
V *' -- 



L-2l --* sn 7r 

it can be shewn as before that the integral is uniformly convergent, and that 
the path can be replaced by a closed contour consisting of the given line and 
an infinite semi-circle to the left of the ?/-axis. The required expression for the 
integral is then obtained by taking the sum of the residues within this contour. 
An exceptional case occurs when a. and j3 are zero or differ by a positive 
integer. Let a.=/3 + 7n, where m is zero or a positive integer; then 
the integrand has poles of the second order at the points -., -oc-1, 
-0.-2, ____ Now the integrand can be written 



so that the residue at the point a. n is 

/ iv n* _ r( 

' 



Hence the integral is equal to 

003 + 1). .. 



.--. 

Let amp = i/r ; then, since - TT < amp ( - ) < TT, it follows that, if < \ff ^ TT, 
mp(-^)=i/r-7r ; while, if > ^ = -TT, amp(-^)=i/r + 7r. 
Accordingly, the analytical continuation of F(a, j8, y, ) when | f | > 1 is 



according as 0<amp^7r or -7r^amp^<0. If OL and ^8 differ by an 
integer, the corresponding changes must be made in the expression. 

If a cross-cut is taken along the real axis from 1 to + oo , the function is 
then uniform in the whole -plane. 

Example 5. Prove 

j* + \og{r(z)}d z =]og^+z \ogz-z. 
If x is real and positive, 



, 
where 0<6'<1. 



154 FUNCTIONS OF A COMPLEX VARIABLE [OH. vm 

Hence * +l \og{T(x)}dx 



where 

= log -s/277 + x log x - x + t(x\ 
where t(x) tends to zero as x tends to infinity. 

Again, / log { F (x} }dx = log x ; 

(jL$Gj& 

so that J log { F (x) } dx = K + x log # - #, 

where K is a constant. 

Thus K must be log >/27r, and therefore 

/ 1 og { F (#)}<&;= log \/27T + .# log #-#. 
J* 

Now the functions on both sides of this equation are holomorphic for all 
values of the variable, provided that a cross-cut is taken along the negative 
real axis from to - x . Hence (55) 

fZ+l I 

I log{F()}c?.s=log\/27r + zlog2 z. 
h 

EXAMPLES VIII. 

1. Shew that 

^4-n...^4.9.-n __ = 2 2 - 1 . 



1.3. 5. ..(2/i- 
2. If 2a=26 (Examples VI. 31), shew that 



Where 



3. Prove that 

4. Prove that 



5. If m is an integer, shew that 



6. If a and 6 are real and >0, and K ()>() shew that 

(i\ 1_ / e i(t>+xi) MX 

(ii) f e-*^**,. .^1 = 0. 



63] 



EXAMPLES VIII 



155 



[In (i) integrate e az z~ n ~ l along x= b, and shew that this path can be 
deformed into that of Example 1, 61 ; in (ii) integrate e-^-"- 1 round 
the contour consisting of x=b and an infinite semi-circle.] 

7. If p is a positive integer and E(w)> - lj shew that 
~ [+> z?(z 2 - I) n dz=2i sin (^ 



where is the initial point, and the initial value of amp(2 2 -l) is TT. 
Deduce that, for all values of n, the integral vanishes when p is odd, and 
that its value when p is even is 



8. . Shew that, for all values of z, 




10. If ft is a positive integer, shew that 



11. If R(2)>0 and amps=\/r, shew that 






Deduce that the expansion is asymptotic if -7r/2 < ^<7r/2. 

[Replace the path of integration by a straight line from O to infinity 
which makes an angle ^ with the positive real axis, and shew that, for 
points on this line, t 2 + l \ i^cos 2 ^.] 

12. If - 7T/2 < amp z=yjs< Tr/2, prove the asymptotic expansion 



f 

Jo 



n-i ..... 

- 1 



where 



| R n | 



13. If s< 1 and -?r/2< amp2=^<7r/2, prove the asymptotic expansion 



where | B. | < 

14. If E(v-oL- / 8)>0, prove 
Bin ra r 



^ \. 
- y + a, 1 - g + a. 1) 



156 FUNCTIONS OF A COMPLEX VARIABLE [OH. 

15. If R(y-o.-/3)>0, prove 

A y, l)-co8,ra^ffiF(a, 1 - 



16. If R(gO>l, prove 



17. If RQo - a) > 0, prove 
EQo-a, g) 



18. If R(p + s) > 0, prove 

' s(s-l) 



19. If = a + ^y, where 5 is a positive constant, shew that the limiting 
value of | F(l +2) | when y tends to + oo is 



20. Shew that the analytical continuation of F(o., /?, y, 1/z) for | z \ < I is 



according as < amp Z^TT or w ~ amp z < 0. 
21. Shew that 



where the integral is taken along a contour similar to that of Example 4, 
63. Values of OL, j8, y, 8, which would make it impossible to draw the 
contour, are excluded. 
22. If RQt?)>0, shew that 



pT(n) 

23. Prove r(z) = ~Limn z B(z, n). 

n *oo 

24. If R(s)>0, shew that 



25. Shew that, if R(s)> - 1, 



and deduce that y = 



vm] EXAMPLES VIII 157 

26. Shew that, if R(z) > 0, R(f ) > 0, 



Deduce f e-' 

Jo 

27. Shew that 



(iii) 
28. Shew that 



r(a)ro8) 



29. If R(y-o.-j8)>0, prove 



30. If a. and /3 are real, shew that 

f r(<x) Infill | 

l|r(oL + l/?)|J nlo I 

31. Shew that, if z+ -1, -2, -3, ..., 



deduce that ^(^) + 7 = 2 - 2 log 2. 

e- u t z - l dt is holomorphic in , and that 

r(e)-e%' T e-*f~*dt 

Jo 
[Integrate e-$ z ~ l round the contour of Fig. 52, 51, and apply the 

inequality r - >- to the circular part of the contour.] 
u TT 

33. If < R(2) < 1, shew that 



34. If < R(s) < 1, shew that 

fcos ^ . ^^ = T(z) cos 

35. If - 1< R(z) < 1, shew that 

jTsin . ^rf<= F(a) sn 

36. If < R(s) < 1, shew that 



158 FUNCTIONS OF A COMPLEX VARIABLE [OH. 



37. If < R(s) < 2, shew that 
'sint , 



38. If B() > 0, shew that 

(i) i= 



(iii) i = (_ 

39. If r > 0, - 7T/2 < $ < 7T/2, shew that 
(i) 

(ii) 0= 



o t 

40. If - TT < < TT, shew that 

t 



(i) log 

(ii) = 



[Put 2 = 1 +e^ e in Example 38, (ii).] 



41. 



42. If R() >- 1, shew that 
r p. I _ /z 

LI ir?* 



l- 



Also, if M is the maximum value of 









for 0^ t^ 1, 



/o 1-* 
Now make ra tend to infinity, and use Example 31.] 

43. If 27^0, -1, -2, ..., prove 



deduce that ^( z ) _ i og g = f (l og ( 1 + J_ ) I ) . 

n =o I \ z + nJ z + n + 1) 

44. If E(g) > - 1, shew that 



vm] EXAMPLES VIII 159 



t_ e -tz\ e -tz_ e -t(z+l)\ / e -t(z+n)_ e -t(z+n+l)\ 






_ -t(z+l) _ -. (z+2) ___ - 






Now make n tend to infinity, and use Example 43.] 

45. Prove ,= 

46. If < R(s) < 1, prove 



ri /- i _ # 2 
^= H^F^J 

Jo 1 ^ 

fr 
47. If E(^)>-1, prove 



_ o/ 1/2 
(ii) 



deduce 






.Apply the transformation l-f. = e T to the third of these integrals, and 
use Ex. 44.] 

48. If H(z) > 0, shew that 



49. If R(^) > - 1, R(z 2 ) > - 1, R^ + 2. 2 ) > - 1, shew that 



50. If 

shew that 






[CH. IX 



CHAPTER IX. 

INTEGRALS OF MEROMORPHIC AND MULTIFORM 
FUNCTIONS : ELLIPTIC INTEGRALS. 

64. Integrals of Meromorphic Functions. If f(z) is holo- 
morphic in a simply-connected region C, F(V)=| f(z)dz is 

Jz 

liolom orphic in that region, provided that the path of integration 
lies entirely within C. If, however, the region C contains one or 
more poles of f(z), the value of F(z) will not necessarily be 
independent of the path of integration, and F(z) may be a multi- 
form function. Each branch of F(Y) will be holomorphic in a 
simply-connected region containing no singularity of f(z). The 
path of integration, of course, must not pass through a singularity 

For example, consider the integral 1 z~ l dz taken along the 




FIG. 66. 



path C of Fig. 66 from 1 to z. This path can be replaced by a 
positive loop from 1 round O and the straight line L from 1 to z. 



64, 65] THE LOGARITHMIC FUNCTION 161 

The integrals along the straight parts of the loop cancel, while 
the circular part gives the value 2-Tri ; hence 



f dz f dz 

-=| 

Jc s JL z 



c L 

Now any path from 1 to z can be replaced by a number of 
positive or negative loops from 1 about O and the line L. Hence 

the most general value of Log z I z" l dz is 

J -f 271? = log z + 2mri, 
L 2 

where ^ is an integer. This agrees with the results of 18. 

Similarly, if a uniform function f(z) has poles a 1? a 2 , ..., of 
residues R lt R 2 , ... , in C, the path from to z can be replaced 
by a series of loops from z about a lt a z , ... , and a straight line L 
from to z. The most general value of the integral will then be 



where 77^, m 2 , ... , are integers. If, however, the residue at the 
pole is zero, the integral round the corresponding loop is zero, so 
that the integral is uniform in the domain of the pole. Thus 



r- 



= z~ l is a meromorphic function throughout the plane. 
Example. Verify, by integrating round suitable loops, that 



where m is an integer. 

65. Integrals of Multiform Functions. If the path of inte- 
gration of a multiform function f(z) does not pass through any 
singularity of f(z), f(z) will vary continuously along the path, 
and the definition of 26 still holds for the integral. As in the 

previous section, the values of F(z)=l f(z)dz may differ with 

J-o 

the path ; and the path can be replaced by a series of loops about 
the singular points, followed by a straight line from z to z. 



]. Let F(s)= / z~ l! -dz, where the initial value of - 1/2 is unity ; 
the integrand has branch-points at the origin and infinity. 



M.K. 



162 



FUNCTIONS OF A COMPLEX VARIABLE [CH. ix 



The loop about 2=00 consists of the line AB (Fig. 67), where A and B are 
the points z = \ and 2 = R (E large) respectively, the circle BCD or U| = R 




FIG. 67. 

described negatively, and the line BA. But this path can be deformed into 
a negative loop from A round O. Hence we need only consider the effect of 
the loops about O. 

Let L denote the positive loop from 1 about O ; then, since Lim z X 2~ 1/2 =0, 

2->0 

the integral round the circular part of L tends to zero with the radius ( 30, 
Th. II.). Also, as z describes the circle, amp z increases by 2?r; so that 
amp2~ 1/2 decreases by TT. Thus z~ l/2 changes from 1/V.r'to 1/V# ; hence 
c dz _ ro dx ri dx . 

JL Jz }\ \/# Jo ^x 

A description of L- 1 , by which we denote the loop L described negatively, 
gives the same result. 

Since z~ lf2 returns to A with the value -1, a second description of L or 
L" 1 will give the value 4, and bring z~ l/2 back to A with the value + 1. 

Thus an even number of loops gives the value 0, and brings z~ l/2 back 
to A with the value + 1 ; while an odd number of loops gives the value - 4, 
and brings z~ l/2 back to A with the value - 1. Hence the general value 
ofF(z)is 2 



where w denotes the integral / z~ 1!2 dz along a straight line from A to z, with 
+ 1 as the initial value of 0~ 1/2 . 



Example**. LetP()= )<&,where/(*)= W(l -2 2 )and/(*)=l initially. 

Also let A and B denote positive loops round the branch points + 1 and - 1 
respectively. 

Since Lim (z-l)-jr- ^ = 0, 

z->i vU-^v 

the value of the integral round A or A" 1 is C, where 



65, 66] THE INVERSE SINE FUNCTION 163 

and f(z) returns to O with the value 1. Two successive integrals round 
A or A" 1 give the value zero, and bring f(z) back to O with the value +1. 
Similarly B or B" 1 gives the integral C, and two successive descriptions 
give the integral zero. Successive descriptions of A and B or of B and A 
give 20 or 2C, while f(z) regains its initial value +1 at O. 

Accordingly, if w denotes / f(z)dz taken along a straight line from O to 2, 

with initial value +1, the general value of ~F(z) is mC + (-l) m w, where m is 
an integer. 

To evaluate C we proceed as follows : make f(z) uniform by a cross-cut 
from - 1 to + 1, and choose the branch of f(z) which has the value +1 at the 
origin on the lower side of the cross-cut. Then, at a point on the #-axis to 
the right of 2=1, amp^/(l 2 2 )^=7r/2 ; so that 

/w- > 



where *J(j? 1) is positive. 



so that //()<&, taken positively round an infinite circle, has the value 2;r. 

But the great circle can be deformed into the loops A and B taken suc- 
cessively, and the value of the integral is then 2C ; hence C = TT. 
Thus the general value of sin" 1 z is given by 




= ( T 

It follows that the inverse function 2 = sin w has the property 
sm{w7r + ( l) m w}=sinw. 

rZ 

Again, since / f(z)dz= -w, it follows that z=ain( w). But z = si 

thus sin( w>)= sin w, so that sinw is an odd function. Many of the other 
properties of the sine function could also be deduced from those of the integral 

/ dz 

J ON /(l-2 2 )' 

66. Legendre's First Normal Elliptic Integral. Let 

= f(z)dz, 



o 

where /(z)= {(1 z 2 )(l k*z z )} 7 , and k is a positive proper 
fraction. The initial value of f(z) at z = is taken to be +1. 
The integrand has four branch-points, +1, 1, +l/& v -^lfc 

The loop A from O about 1 gives the integral 2K, where 

Ii fl x 

o S /{(1 -a 2 )(l-fe 2 )}' and f^ returns to with the value 
1. Two successive integrations round A give the value 0, and 



164 



FUNCTIONS OF A COMPLEX VARIABLE [en. ix 



bring f(z) back to O with the value 1. Similarly the loop B 
about 1 gives the integral 2K, and two successive integrations 
round B give the value 0. Successive integrations round A and 
B or round B and A give the values 4K and 4K respectively, 
and f(z) regains the value + 1 at O. 

Since a straight line cannot be drawn from O to l/k without 
passing through the singularity +1, the loop L x about l/k is 
formed by means of a curved line (Fig. 68) above the a?-axis and 




FIG. 68. 



a small circle about l/k. This loop can be deformed into the 
contour (Fig. 69) consisting of : 
(i) the ic-axis from O to 1 e y 
(ii) a small semi-circle c of centre 1 and radius e above the 

a>axis, described negatively ; 
(iii) the ic-axis from 1+e to 1/k e; 
(iv) a small circle C of centre l/k and radius e, described 

positively ; 

(v) the ic-axis from I/A; e to 1 + e; 
(vi) the semi-circle c described positively ; 
(vii) the ce-axis from 1 6 to O. 




FIG. 69. 



Since Lim (z l)f(z) = and Lim (z l/k)f(z) = 0, the integrals 

->! s->lfk 

along (ii), (iv), and (vi) tend to zero with e. 

The integral along (i) gives K. As z passes round c, amp (z 1) 



66] 



LEGENDRE'S FIRST NORMAL INTEGRAL 



165 



decreases by x, and (1 x) changes to (x l)e~ iir ; hence the 
integral along (iii) is 

CtX . -r-r f 



1 

where 



TT< 

K = 






Again, as z passes round C, amp (z 1 /k) increases by 2x, and 
(1 kx) becomes (1 kx)e 2iir ; hence (v) gives the integral 

dx 

jW* 



Finally, as z passes round c, amp (z 1) increases by x, and 
(x 1) becomes (1 x)e iir ; so that (vii) gives the integral 

dx =K _ 



Thus the value of the integral round the loop is,2K+2iK', 
and/(;s) returns to O with the value 1. 

It can be proved in a similar manner that the integral round 
the loop L 2 (Fig. 70) is 2K 2^K'. This follows more simply, 



Y, 




FIG. 70. 



however, from the fact that L 2 can be replaced by A, L 1? A' 1 , 
taken in succession : the value of the integral along this contour 
is then 2K-(2K + 2iK / ) + 2K = 2K-2iK / . 

Similarly, the contour C x (Fig. 70) can be replaced by A and Lj 
taken in succession ; so that the integral round C^ has the value 
2K-(2K+2iU / )=-2^K / , and /(z) returns to O with the value +1. 

Finally, the integrals round the loop L 3 and the curve C 2 have 
the values -(2K + 2iK / ) and 2iK' respectively. 

Hence, if w denotes the integral || f(z)dz taken along a 

Jo 

straight line from O to z, with the initial value + 1 at O, the 



166 



FUNCTIONS OF A COMPLEX VARIABLE [CH. ix 



general value of F(z) is 2mK + 2niK'+( I) m w, where m and n 
are integers. 

The value of the integral when z is infinite can be found 
as follows. Let the integral be taken round the contour (Fig. 71) 




consisting of : (i) the straight line from O to z ; (ii) a semi-circle 
of centre O from z to z ; (iii) the line from z to O. Since 
this contour is equivalent to the contour C 2 (Fig. 70), the integral 
has the value 2iK'. But the integral along (ii) tends to zero 

when z tends to infinity (30, Th. I.), and [ f(z)dz= [* f(z)dz, 

J -z Jo 

since the final value of /(z) is equal to its initial value. Therefore, 

when z tends to infinity, I f(z)dz tends to the value -&K'; so that 

Jo 



f(z)dz = i 



If in the integral 



fl/A 

Jl * 



dx 



we put y=-^/(\kV)/k', where k' = *J(l k 2 ), we obtain 

K--P dy 

Jo 



It follows that K' is the same function of k' that K is of k. 

Inversion of the Elliptic Integral. In Example 2 of the 
previous section we deduced from the properties of the integral 

w = 1 -^ ^\ various properties of the inverse function z = sin w. 



o 



66, 67] INVERSION OF THE ELLIPTIC INTEGRAL 167 

Similarly, if iv=\ /f/1 2\?i 72 2\v z can ^ e regarded as a 

Jo v I \ ^ / \ ^ / / 

function of w, and from the properties of the integral those 
of the function can be deduced. We shall here make two 
assumptions : (i) that the function exists for all real or complex 
values of w ; and (ii) that the function is single- valued. These 
assumptions will be justified in Chapter XL The function is 
denoted by z = saw: from the general value of the integral it 
follows that 



Accordingly, sn w has two periods, 4K and 2^K', the one purely^ ^ 
real and the other purely imaginary, and sn(2K w) = sn^. -" 



Again, since f(z)dz= \ f(z)dz= w, it follows that 
Jo Jo 

z = sn ( w) = sn w ; so that sn w is odd. The properties of 
the integral also give : 

snO = 0, snK = l, 



Instead of sn w the notation sn (w, k) is frequently employed : 
k is called the Modulus and k' the Complementary Modulus 
of sn (w, k). 

Example. Shew that K' = log (4/&) + <(&), where <j>(k) tends to zero with L 
We have 



-f ( - 

J* WO-# 2 ) 

where y = kx. 
Hence 




= log2; 
from which the required theorem follows. 



67. The Weierstrassian Elliptic Integral. Let 

t(;-l^ = f(z)dz, 

J ZQ 

^liQref(z)={4>(z e l )(z e 2 )(z e s )}~^: here w = w corresponds 
to Z = Z Q) and one of the two values of f(z ) is selected as initial 



168 



FUNCTIONS OF A COMPLEX VARIABLE [CH. 



value. There are four branch-points of f(z) (Fig. 72), e l} e, 2 , e 3 , 
and oc . The loop L about oo , however, consisting of the line from 







z to f and a large circle described negatively, can be replaced by 
the loops L x , L 2 , L 3 , about e lt e 2 , e 3 , described negatively in suc- 
cession ; so that it is only necessary to consider the effects of these 

three loops. Let A 1= f(z)dz t A 2 = f(z)dz, A 3 = f(z)dz\ 

Jzo **9 ^ Z 

then integrals round the loops L u L 2 , L 3 , or Lf 1 , L.J 1 , L^ 1 , give 
the values 2A 15 2A 2 , 2A 3 , respectively. Two successive integra- 
tions round a loop give the value zero. Successive integrations 
round loops L,, and L 6 . give the value 2A r 2A S . Again, the 
description of an even number of loops brings f(z) back to z 
with its initial value f(z ), while an odd number brings it back 
with the value f(z Q ). 

Hence, if I denotes the integral I f(z)dz taken along a straight 
line from z to z, the general value of the integral is given by 



^XXq \~ \ / ' 

where n lt n 2 , n^ are integers such that ?i x -f ii 2 + % has the value 
or 1 according as the number of loops described is even or 
odd. 

Now let n^ in 2 , n s = w\, so that either / n 2 = m 1 + m 2 or 
n% = ?7i 1 + m 2 -f 1 ; then either 

w-w = -2^- A 2 )-2m 1 (.A 3 - A 2 ) + I 



67, 68] THE WEIERSTRASSIAN ELLIPTIC INTEGRAL 169 

|% re, 

Again, if Wi=\ f(z)dz and oo. 2 = I f(z)dz, 

JeS J <>! 

A i A A r$\ 

A 2 A 3 = w 1 and A 2 A 1 = o> 2 ; 
hence either i# = u> +2??i 1 ft) 1 + 2??i 2 ft> 2 + I 



or w = 

Thus the inverse function z = (j>(w) is doubly-periodic, with 
periods 2co 1 and 2co 2 . 

Next, let the integral be taken along the contour consisting of 
the loops L, L 3 , L 2 , L 1? taken in succession. This curve encloses 
no singular point, so that the value of the integral is zero. But 
the integral round the large circle tends to zero as the radius 
tends to infinity ; hence 

= 2 /0)<i2;-2A 3 +2A 2 -2A 1 ; 
so that f f(z)dz = A 3 - A 2 + A r 

J*o 

Now take w = \ f(z)dz= A 3 -f A 2 A x ; then w= I f(z)dz. 

J t J CO 

Hence, if z = e lt 

w = 2??!^ + 2?^ 2 ft) 2 A 3 + A 2 Aj + Aj = < 2m jr w 1 + 2m. 7 w 2 + ^ 
or w = 2m t co x + 277i 2 w 2 A 3 + A 2 A 1 + 2 A 2 A x 
= 2?^ ^ + 2m 2 ft> 2 + 2o> 2 + w 1 . 

Therefore ^ 1 = 0(ft> 1 ). Similarly 6? 2 = 0(ft) 1 + ft) 2 ) and e 2 = <p(a). 2 ). 
Again, if W = ty H- 1, . 

te; = 2m x &>! -f 2m 2 w 2 + W 
or w = 2m 



Thus <f>(iv) is an even function of iv. It will be shewn in 
Chapter X. that <f>(w) is Weierstrass's Elliptic Function ^(w). 
It should be noticed that the signs of the two periods 2co l and 
2o> 2 depend on the initial value selected for f(z ). 

68. Elliptic Integrals in General. Any integral of the type 
JR(z, Jfydz, where R(#, y) is a rational function of x and y and 

Z is a polynomial of the third or fourth degree in z with real 
coefficients and no repeated factors, is called an Elliptic Integral. 



170 FUNCTIONS OF A COMPLEX VARIABLE [CH. ix 

When Z is a cubic. the integral can be transformed into an 
integral in which Z is a quartic as follows. 

Let Z = (2 /3)(az* + bz + c), where /5, a, 6, c, are real; then, 
if *-=, 



which is an integral of the required form. 

Again, let R(#, y) = P(x, y)/Q,(x, y), where P(#, y) and Q(&, y) 
are polynomials in x and y ; then, since (\/Z) 2 ^, where p is a 
positive integer, is a polynomial in z, we can write 



where K(z), L(z), M(z), N(z), are polynomials in z. 
Now multiply numerator and denominator by M(z) 



where U(2) and V(2f) are rational in z. 

But U(2;) can be integrated by elementary methods. Hence 
we need only consider integrals of the type 



or 



where S (2) is rational in 2. 

Again, by the method of partial fractions, S(0) can be put 
in the form 



Hence the integral J{S(2;)/\/Z}cZ2; can be expressed linearly 
in terms of integrals of the types 



z n -, , dz 

dz and 



Now ( 

f^n 
:= dz can be expressed in terms of the four integrals 

f* 3 7 f* 2 7 f* ,7 f^ 2 

JTI^ J^* JTI^ J72- 

But 



68] REDUCTION OF ELLIPTIC INTEGRALS 171 

where Z = az' i -}-bz s +cz 2 +dz + e; therefore 1 T^- can be expressed 
in terms of the three integrals 

[z 2 dz [zdz [dz 

1*7 ' \ /7 ' 1/7' 

J V " J \ J J V " 

Similarly, since 



dz (Z - OL) m ~ (z-OL) m + l v/Z 

j= can be expressed in terms of 

JO-afv/Z 

f dz Ccte C(z-oi)dz Kz-a.) z dz 

JO-(x)v/Z' JN/Z' J \'Z J VZ 

Thus every Elliptic Integral can be expressed in terms of 
integrals of the types 

dz 



dz Czdz Cz 2 dz 

7z' J7z' J7I"' 



Again, since imaginary factors of Z always occur in pairs, 
Z can always be written a(z 2 +pz + q)(z 2 +rz + s), where p, q, r, 
s, are real. Now in the transformation z = (/-f<7 )/(! + )> ^ 
f and g be chosen so that the coefficient of in each quadratic 
is zero ; then Z will take the form 



It is always possible to find real values for in and n. For 

^. n Q s * ps or 

f+q = 2+ - , fq = - -- ; 

r-p' J * r-p ' 

so that / and g are the two roots of the quadratic equation 

O - P)f 2 + 2 (s - #)/+ (ps - gr) = 0. 
Accordingly, if the roots are real, we must have 

(s-q) 2 -(r-p)(ps-qr)>0. (A) 

Now let the two equations 

x*+px + q = Q, x 2 +rx + s = 0, (B) 

have roots x : , x z , and x s , x, respectively; so that 



Then inequality (A) can be written 
(x l - a; 3 ) (x 1 - aj 4 ) (x. 2 - 



172 FUNCTIONS OF A COMPLEX VARIABLE [OH. ix 

This inequality holds if one at least of equations (B) has 
imaginary roots ; for then the four factors consist of two pairs 
of conjugate complex quantities. Also, if both equations have 
real roots, the factors of Z can always be chosen so that 



Thus the inequality holds in this case also. It follows that 
real values of / and g, and therefore of m and n, can always be 
found. 

Accordingly, every Elliptic Integral can be expressed in 
terms of integrals of the types 



where Q = 

But 
and this integral can be evaluated by elementary methods. 

Also 



d . , f d? 

-ifl w->vo' 



and the last integral can be evaluated by elementary methods. 
Hence we need only consider the integrals 



There are four cases to be considered (we assume a 



In case (i) put =x/a, k = b/a', then the integrals are trans 
formed into integrals of the forms 

x*dx f dx 



fc^ 



dx 



In cases (ii), (iii), and (iv), make the substitutions 
l-a 2 f 2 = a 2 , l-W^ = x, and l + fe 2 f 2 = 
respectively ; then all these cases reduce to case (i). 



68, 69] COMPLETE ELLIPTIC INTEGRALS 173 

dx 



Now 

JVUi-^Xi-/^)} A^JVUI- 

| r //I _ 7.22 



dx. 

" ""' ic 



Hence all Elliptic Integrals can be expressed in terms of Ellipti 
Integrals of the three types, 

dx 



The three definite interals, 



f* 

Jo(^ 2 - 



dx 



are called Legendre's Normal Integrals of the First, Second, and 
Third kinds. 

Example. Prove 

f 1 3^ + 2^ 2 , _ /r ? / 

Jo v^+^+i 1 * 3jo 
- 



But 
P ' dx 

Jo ^PT 



' where ^ 

2 /vs/s ^ 3 

3 Jo 



3 
Hence the required equation follows. 

69. Complete Elliptic Integrals. If in the First and Second 
of Legendre's Normal Integrals the substitution x = sin is made, 
they become 



174 FUNCTIONS OF A COMPLEX VARIABLE [CH. ix 

respectively. In particular, if x = l, then <f> = ir/2, and these 
integrals become 



E = E(&, 7T/2) = a - A? sm*<p)dfa 

Jo 

which are known as Legendre's Complete Elliptic Integrals of 
the First and Second kinds. Similarly we write 
K' = F(F, 7T/2), E' = E(#, 7T/2). 

These functions can be expressed as hypergeometric series in 
k and k' : for, since k < 1, 



^^ 



Similarly K' = F(J, }, 1, 7c' 2 ), E = -F(- J, J, 1, A; 2 ), 

2j & 

The numerical values of K, E, K', and E' can be easily evaluated 
by means of these series, except when the value of k or k', as the 
case may be, is nearly unity, in which case the convergence is 
siow. 

Landens Transformation. If in the integral F(k, 0) we 
make the substitution 

tan(0 1 0) = &'tan0 or tan 1 = sin 20/(& 1 + cos 

, 7 1 k' k 7^7 T , . 

where A; 1 = -._, y / = /1 7 / V ;^<C^> we obtain 
l + /c ( 1 + /c )" 



and 



so that Y(k, 0) = 

Thus the integral is expressed in terms of an integral of 
smaller modulus. In particular, if = Tr/2, then fa = TT, so that 



69, 70] LANDEN'S TRANSFORMATION 175 

Accordingly, if the modulus of k is nearly unity, the value of 
F(k, 7T/2) can be deduced from that of F(/<; 1 , 7r/2) by means of 
this transformation. 



Bmmpl e. Prove 



and deduce from the example of 68 that 

P3.^+2^^. -1/1 TT 

Jo 



70. Legendre's Relation. A relation can be established 
between the four quantities K, K', E, E', as follows. 
We have 



dK_W* ksm*<}>d<j> _1W 2 d<f> _K 

dk ~ J o ( 1 - /c 2 sin 2 0) 3 /' 2 ~ k] ( 1 - /c 2 sin 2 ^) 3 / 2 k ' 

1z d sin0cos0 7c' 2 . 79 . 

But k 2 ~j- -rrtj r, . y = 7^ 79 . 2j vo/. 2 + v/( 1 & 2 sm 2 0). 
2 2 23 / 2 

Th, fore 



Hence ^^Jl _f C'/M &M tt & 

Accordingly, since 7c 2 + 7c' 2 = l, 

rfK E VK ,'ti 

, , , = T77T-/ + -TIT A JT^ ^ 



Therefore, interchanging 7^ and /c', we have 
Again, 




dK' = _E' + &K; //- 



1 f-/ 2 7/1 1 f ff / 2 d0 E-K 

J(l k-sm 2 d))d<j> f . = - 

/.'J /^Jo l A; 2 sm 2 / 



,_ K '>. 

CvK 1C 

f /\Y 
Accordingly, if W = KE r -h K'E - KIv, -^- = ; therefore \V is 

a constant. 



176 FUNCTIONS OF A COMPLEX VARIABLE [OH. ix 

Now consider the value of (E K)K' when k tends to zero. 



Also 



Hence |(E- K)K' < (^& 2 +. 
so that Lim {(E-K)K'}=0. 

But, when fc = 0, K = 7r/2, and E' = l ; therefore 

W=KE' + K'E-KK' = |. 
COROLLARY. K and K' satisfy the equation 



where x = k 2 . This equation is known as the differential equation 
of the Quarter Periods of the Jacobian Elliptic Functions. 

EXAMPLES IX. 

rZ 7 

1. If w= I 4> and if w is any value of w corresponding to z=z 0t 

shew that the general value of w for z=z is w +m>/2ir/4-f'N^7rt/4, where 
m and n are integers, such that m + n is even. 

ri +s 2 
3<5?2, shew that, with the notation of the previous example, 

the general value of w is w -H773 + mr\/3/3, where m + n is even. 

3. Find the most general value of / f , for any path of integration, 

JO ^(Z' + I) 

where the initial value of the integrand is unity. 

Am. wn + (-l) n log(l+\/2), .(w = 0,. 1, 2, ...). 



5. Prove that, for the ellipse # 2 /a 2 +y 2 /6 2 =l, the length of an arc 
measured from the point (0, 6) in the clockwise direction is aE(e, <), where 
e is the eccentricity and ^'=asin <. 



6. Prove that [Vws".r^=2v/2E(4=' 

\2 



where cos ^ = cos 2 



70] EXAMPLES IX 177 

7. If a 1 > b- > c 2 , shew that 

y" 00 f^A 2 

'- + A)}~v'O 2 - c2 ) 



8. Shew that 



Ji 



where = sin 15. 

[Shew that the integral is equal to 

2 r x o?y 

where A = (l - 



9. Shew that 

where ^=cos!5. 
10. Prove 



f flLy 2 / . TT\ 

- =Fsml5 ' 



11. By means of the substitution ./; = (4 - T/ 3 )^ 2 , shew that 

dy . 



deduce that K' = V3K, where /fc=sin 15. 

^ Stan 3 + 8 tan 2 # -2 tan 0+4 
A H,, V(l + 2sin2^) - 

B 



prove that 



13. Prove that the length of the lemniscate r = a\fcos20 is 2\/2F( - , J ) 

\V2 / 

14. If -s' denotes the length of an arc of the hyperbola x* i \a, i y 1 \l> 1 =\ 
measured from the point where it crosses the ^-axis, shew that 



where J 

15. Shew that, if K = ^ and K' = /'-, 



''K liKK (*K ZK 

I 'rove tliat (K K'!\) satisfies the differential equation 

4KK 1 -j^-_ t y. 

M.F. M 



178 FUNCTIONS OF A COMPLEX VARIABLE [OH. ix 

16. Shew that, if n>l, 

(i) n T k n Wdk=(n - 1) T "- 2 E'<tt ; 
Jo Jo 

(ii) (n + 2) r y^EW/j = (n + 1) T L M K'dL 

Jo Jo 



17. If P is any point on that branch of. the hyperbola .r-/a 2 - 
which crosses the #-axis at A, shew that the difference between the arc AP 
and the portion of the asymptote cut off by a perpendicular on it from 
P tends to the limit 



as P tends to infinity. [Cf. Example 14.] 

18. Shew that 

Y 2 dx _ _ * 

J* V{(i-^)(^' 2 +^ 2 )}" 

where ^= x /(l-^7 2 ). 

19. Shew that 

dy 



where ky = *J\-x z . 



CH. X. 71] 



CHAPTER X. 
WEIERSTRASSIAN ELLIPTIC FUNCTIONS. 

71. Doubly-Periodic Functions. A uniform function F(z) 
which has two primitive periods 2 and Q' is said to be Doubly- 
Periodic. For all values of z, 



so that 

where m and m can have any integral values. 

THEOREM. The two primitive periods Q and Q' cannot have 
the same amplitude. 

For, if they have the same amplitude, let Q = pe ie , Q' ==//**, 
and assume /o>/o'. Then, if tt" = tt-Q' = (p-p)e ie , IT is a 
period of modulus less than p. Let this process be repeated 
with the two periods Q' and Q" '; and so on. After a sufficient 
number of steps a period is obtained either of modulus zero 
or of modulus less than any assigned quantity. 

The first case cannot occur, however; for if u> denote the 
value of the two equal periods subtracted in the last step of 
the process, 



where p and q are integers ; but this is impossible, since Q and 
Q' are primitive periods. 

In the second case, if denote the period, the function 
{F(z) F(z )} has zeros at z and z +et>. Accordingly, F(0) has 
essential singularities at all points of the plane ( 22, Theorem I. 
Corollary 1). Such functions are excluded from consideration. 

Congruent Points. The points z + mil + m'l', where in and 
IK' may have any integral values, are said to be congruent to 
the point z. 

/'< ,'iod- Parallelograms. A parallelogram of vertices a, a + Q, 
a 4- 2', <t + fi + !Y, is called a period-parallelogram. It is sufficient 



180 FUNCTIONS OF A COMPLEX VARIABLE [CH. x 

to study the behaviour of the function in one period-parallelo- 
gram in order to know its properties for the entire z-plane. If 
the whole plane be divided up by two sets of equi-distant 
parallel lines into a net-work of period-parallelograms, corre- 
sponding points of the parallelograms form a set of congruent 
points. An example of such a net- work was given in 37. 

72. Elliptic Functions. A doubly-periodic function with no 
singularities in the period-parallelogram except isolated poles is 
called an Elliptic Function. It is convenient to choose the 
periods 2^ and 2o> 2 so that, as in 37, I^/o^) is- positive. 

Weierstrass's Elliptic Function. If we differentiate the series 



Z) = -s 




we obtain 

From this series the equations 

?'(z + 2 Wl ) = p'CO, p\z + 2o> 2 ) = v'(z\ 
follow immediately; so that <@>(z) is an Elliptic Function. 
Again, integrating, we have 



Now let z = W then 



so that C = 0. Thus #> (z + 2^) = p(z). 
Similarly %>(%+ 2o> 2 ) = p(2;). 

Accordingly, p(z) is an Elliptic Function. 

COROLLAKY. If n is any integer, {$>(z)} n is an elliptic Junction. 
Note. The notation <p(z\ co 1 , w 2 ) is sometimes used instead of 

P(*> 

THEOREM I. The derivatives of an elliptic function are 
elliptic functions. 

For, if f(z + 2^) =f(z), f(z + 2w 2 ) =f(z), 

it follows that 



THEOREM II. An elliptic function must have at least one 
pole in a period-parallelogram. 



71,72] WEIERSTRASS'S ELLIPTIC FUNCTION 181 

For if not, the function would be finite at every point of the 
plane, and would therefore, by Liouville's Theorem, be a constant. 

Thus the function $(z) has poles of the second order at the origin 
and congruent points ; while at all other points it is holomorphic. 
The principal part at the origin is 1/z 2 . Similarly p'(z) has a 
pole of the third order at the origin, with principal part 2/z 3 . 

COROLLARY. If two elliptic functions have the same periods 
and the same poles, and if their principal parts at the poles are 
equal, they can only differ by a constant. 

Note. An elliptic function has an essential singularity at 
infinity : for it has an infinite number of poles in any neighbour- 
hood of infinity (cf . 48, Note). This holds true for all periodic 
functions ; e.g. cot z. 

THEOREM III. An elliptic function can have only a finite 
number of poles in a period-parallelogram ( 22, Theorem 2). 

THEOREM IV. The sum of the residues of an elliptic function 
f(z) at points in. a period-parallelogram is zero. 

Let y denote the parallelogram ABCD (Fig. 73) of vertices 




Fio. 73. 



((, r/,H-2o) 1 , a -f 2^ + 2o> 2 , a+2a> 2 , drawn so that none of its sides 
passes through a singularity of f(z). Then the sum of the 
residues of f(z) in y is given by 



a+2o> 2 



= 0. 
For example, the residues of p(z) and $(z) at 2 = are zero. 

COROLLARY. An elliptic function cannot have a single simple 
pole in a period-parallelogram. 

Oi'der of an Elliptic Function. The number of poles of an 
elliptic function in a period-parallelogram, a pole of order s 



/ 



182 FUNCTIONS OF A COMPLEX VARIABLE [OH. x 

being counted as s poles, is called the Order of the function. 

It follows from Theorem IV. Corollary, that the order of an 

elliptic function must be not less than 2. 

The two simplest types of elliptic functions are : 

(i) functions with a single pole of order 2, at which the 

principal part is of the form A/(z-oc) 2 , in each period-parallelo- 

gram ; q>(z) is a function of this type : 

(ii) functions with two simple poles of principal garts 

A/(z a) and A./(z (3) in each period-parallelogram ; it will be 

shewn in Chapter XL that the Jacobian functions snu, cnu, 

dn u, are of this type. 

THEOREM V. The number of zeros of an elliptic function f(z) 
in a period-parallelogram, where a zero of order r is counted 
as r zeros, is equal to the order N of f(z). 

For (31, Corollary 1) 



where y denotes a period-parallelogram. But, since f'(z)/f(z) is 
an elliptic function, this integral is zero (Theorem IV.). Hence 

Sr = 2s = N. 

Thus, since p'(z) has one pole of order 3 in the period- 
parallelogram, it must have three and only three zeros in 
the parallelogram. Now, substituting z = ^ in the equation 
$(z + 2ft*!) = $>'(z), we obtain ^'(^i) = %>'( o^). But from the series 
for $>'(z) it follows that %>'(z) is odd : hence ^'(^i) = 0- Similarly 
/(o> 2 ) = 0, ^(co l -\-(a z ) = 0. Thus the only non-congruent zeros of 
p'(z) are co lt w 2 , and o^ + o^. 

COROLLARY. Since the elliptic function {/(z) C} has the same 
- poles as f(z). the number of its zeros in a period-parallelogram 
will be N. Hence the number of points in a period-parallelogram 
at which /(z) = C is N. 

THEOREM VI. If the elliptic function f(z) has p zeros a 1? 
a 2 , ..., dp, of orders r l9 r 2 , ..., r pt and q poles 6 1? 6 2 , ..., b q , of 
orders 8 lt s 2 , ..., s q , in a period-parallelogram, 



m=l 7i 

where X and /x are integers. 



72, 73] POLES AND ZEROS 183 

/\ 

For, if y denote a period-parallelogram (31, Corollary 2), 



= - 2o> 2 Log 1 -f 2^ Log 1 



Hence 2 r a 

l = l 

Example. Prove that ^= ^ irisa simple zero of 



This is an elliptic function in u of order 3, its only pole being at u=Q. 
Two zeros are u = v and tt, = w, so that the third must be congruent to -v w 
(Theorem VI.). Also (Theorem V.) each zero must be of the first order. 

73. Relation between $(z) and p'(z). We shall now prove 
that $(z) satisfies the differential equation 



where # 2 and </ 3 are constants. 
Near z = we have 



( (z- 
Accordingly 

/ A - 

= 



But if ii is odd, 22 = ' therefore 



where fc- 

' = 



184 FUNCTIONS OF A COMPLEX VARIABLE [CH. x 

From this equation we derive the following equations : 



Hence, if <j>(z) denotes the function 



near 2 = 0, cj>(z) = Dz 2 + E,2 4 + .... 

Thus the elliptic function <f>(z) has no pole at the origin. But 
the origin is its only possible pole. It is therefore a constant 
(Theorem II. 72) ; and since 0(0) = 0, the constant is zero. Thus 



The quantities g 2 and g 3 are called the invariants of #?(z). It 
sometimes found useful to use the notation $(z ; g^g^) fo 

COROLLARY. By differentiating equation (A) we obtain : 



Thus every derivative of <p(z) can be expressed as a polynomial 
in z and 'z. 



Example. Prove that the function {jp(u)fl(u) + $P(u) 1} has five zeros, 

r=5 

w i> U 2t U si u i u &> i n a period-parallelogram, sucli that S^-^SAwj + S/zwo, 

r=l 

where A and /x are integers. Verify that, if 2 = >(), these values of u give 
the five roots of the equation 



If /(z) = 0, equation (A) becomes 



Now we know (Theorem V. 72) that 0>'( w i) ^'(^2)' P / ( w i 
are all zero. Hence the three roots of this cubic in p(z) are e lt 
e 2 , 6 3 , where *e 1 = P(i) ^ = ^(^1 



73, 74] THE ADDITION THEOREM 185 

It follows that equation (A) can be written 



If the coefficients in equations (A) and (B) are equated, the 
following important relations are obtained : 



The Weierstrassian Elliptic Integral. Let z $>(w) : then, since 




Now when w = 0, z = cc ; therefore 
w= I 

J r 



The two branches of the integrand give equal and opposite 
values of w, which correspond to the same value of z, since 
is even. 



74. The Addition Theorem. Consider the elliptic function 



The functions p(u + v), $>(u), and p'(u) have poles at u= v. 
u = 0, and w = respectively; while {p(u) <p(v)} has zeros at 
u=-v. Hence the only possible non-congruent infinities of 
f(u) are -it = 0, u= v. 

Near u = (), 




Accordingly, when u = 0, f(u) is finite and has the vahu- 
zero. 



186 FUNCTIONS OF A COMPLEX VARIABLE [CH. x 

Again, let u = v + e ; then 



Hence /( 16) is finite when u = v. 
Finally, let u = v + e ; then 



Hence /(u) is finite at u= v. 

Thus f(u) is constant (Theorem II. 72). But when u = 0, 
f(u) has the value zero ; therefore 



This is the Addition Theorem for the Weierstrassian Elliptic 
Function. 

COROLLAKY. p(u-v)= -y 



Prove 
Duplication Formula. If u = -v + e, the addition theorem gives 



74, 75] ADDITION OF A SEMI-PERIOD 187 

Therefore, if e = 0, 



Example. Shew that 



The following three formulae can be deduced from the 
addition theorem : 





The proof is left as an exercise to the reader. 
Example. Prove 



75. Properties of the Zeta Function. Integrating the equa- 
tion 



we have f (u + 2^) = f (w) + 2 

where 2^ is a constant. 

Now, let u = - w l ; then f (o^) = 
so that ^i = 

Similarly f (t6 + 2o, 2 ) = % , 

where /7 2 = f (o) 2 ). 

It follows that 

f (u + 2???.^ + 2^w 2 ) = 
and that f ( mo^ + 7io) 9 ) = 

The Zeta function is not an elliptic function. It possesses, 
however, a sort of periodicity, and is called a Periodic Function 
of the Second Kind. In each period-parallelogram it has a 
simple pole congruent to 16 = 0. The residue at this pole is 
unity ; for, if we integrate 



we obtain 



188 FUNCTIONS OF A COMPLEX VARIABLE [CH. x 

But, since g(u) is odd, C = ; therefore 



Example. Shew that f (2w) = 2f (u) + X ^' j& . 

2 <? vv 

Again, let f (u) be integrated round the period-parallelogram y 
(Fig. 73) ; then 

r ra+2a>! |*a+2a> 2 

I ^(^)^^ == l {(^) (^ ~l~ 2ft> )}cfat I {(^0 

J v J a J a 



= 2-Tri, 
since there is only one pole in y. Thus 



This is Legendre's Relation for the Weierstrassian Elliptic 
Functions. 

THEOREM. Any elliptic function can be expressed linearly in 
terms of zeta functions and the derivatives of zeta functions. 

Let f(u) be an elliptic function of periods 2^ and 2o> 2 , and 
let a, 6, c, . . . , & be its poles in a period-parallelogram. Also let 
the principal parts of f(u) at these poles be 

_Ai _L ^2 , , A"i 

u - a "*" (u - a) 2 "*" ^ (w - .)"i ' 

B~D "D 

i JD O D. 



U /C (U , 

Then consider the function 



75, 76] THE ZETA FUNCTION 189 

This function is finite at all points of the period-parallelogram. 

Also 



= 0(w). (Theorem IV. 72.) 

Similarly <j)(u + 2a> 2 ) = 0(u). 

Accordingly, <j>(u) is a constant (Theorem II. 72); therefore 



(A) 



(74 1)! 

Example. Shew that 

2f (2w) + 2r h + 2r? 2 = () -4- f (w + CD,) + {(u + w, + o> 2 ) + f (w + o> 2 ). 



76. Properties of the Sigma Function. Integrating 



we have log{<r(u-h 2^)} =log{<r(t6)} + 2 % u + C; (cf.50) 

or o-(u + 2w 1 ) = CV(u)6 2 ''i it . * 

Now let u = w l ; then o-(fo) 1 ) = CV( a) l )e~ 27? i a) i, 
so that C'= -e 2l ii. 

Therefore o-( w + 2^) = 

Similarly <r(u + 2o> 2 ) = 

By the method of induction it can be deduced that 
ar(u + 2771^ + 27K 2 ) = ( l) wl '' l + 

The Sigma function is called a Periodic Function of the 
Third Kind. 

Near u = Q we have 



Ih-nce 

But Lim {-^ [ = 1, so that C = ; therefore 

->0 v * J 



Thus 



190 FUNCTIONS OF A COMPLEX VARIABLE [CH. x 

THEOREM. Any elliptic function can be expressed in terms of 
sigma functions. 

Let /(it) denote an elliptic function of periods 2u> lt 2a> 2 , having 
in a particular period-parallelogram zeros a lt a 2 , ..., a pt of orders 
m lf m 2 , ..., m p , and poles b lt b. 2 , ..., b q , of orders %, 7i 2 , ..., n q . 
Then consider the function 



We choose the a's and fe's so that 2?rfcct 2^6 = 0, replacing, if 

necessary, some of them by congruent points (Theorem VI. 72). 

Now (it) is finite at all points of the period-parallelogram. But 



(Theorem V. 72.) 

Similarly </>(u+ 2o> 2 ) = 0(u). 

Thus (Theorem II., 72), 0(u) is a constant ; so that 

*/ \ / I ./ /i , 



For example, the function {^(^) ^(^)| has two simple zeros 
t>, and a pole of order 2 at u = 0; therefore 



In this equation let u be small ; then 



Hence, equating the coefficients of ,, we have 

TV 




so that P( tt )-P< ff )=- 



COROLLARY. If in equation (A) we put v = it + e> and make 
tend to zero, we obtain 



Example 1. Shew that 
n'f \ 



76] THE SIGMA FUNCTION 191 

Again, if equation (A) be differentiated logarithmically, 



In this equation interchange u and v ; then 

' - {(u - v) - 2( v). 



Hence - = u + v - u - v . (B) 

2 



COROLLARY. If in formula (A) of -75 we make the substitution 



2 #>(u) 

and similar substitutions for (u b), ..., f(t& &); then, since 
2A 1 = 0, it follows that f(u) can be expressed as a rational 
function of >u and >'u. 



Example 2. From equation (B) deduce the addition theorem 



EXAMPLES X. 

1. Find, the zeros of 



and shew that they are all simple zeros. Ans. v r w. 

2. Find the poles and zeros of 



Ans. Simple poles, v, w\ simple zeros, 0, v-w. 

3. If %>(z) is constructed with 2<it l9 2w 2 , as primitive periods, while $>i(z) 
is similarly constructed with 2d> 1 /72,.2(o 2} as primitive periods, prove that 



4. Shew that V 

+ ( 



(i) 
(ii) ^ 

5. Shew that 
(i) 



192 



FUNCTIONS OF A COMPLEX VARIABLE 



[CH. 



6. Prove 

7. Shew that 



8. Prove ff>(2y)-p(2w)=- 

9. Shew that 



(ii) 

10. 



. 






, 

~ 



11. Prove 

12. Shew that 

i 

13. If *()=f 

and ^(^ = 
shew that 

14. Shew that 






15. Prove 

16. Prove 

(r(a + b)(r(a b)(r(a 

17. Shew that ? 

(7 

18. Shew that a-(2u) 

19. Prove 



- d) - v(a + c)v(a - c)v(b + d)<r(b - d) 



u ^( u -<tfiMM- Q) 2 )<r(m-<u 1 + u> 2 ) 
o-(w 1 )cr((o 2 )fr((o 1 -f w 2 ) 



= 2- 



(0 - w)<r(w - u)<r 



tr 3 (?*) (T 3 (v) or 3 ( M; 



EXAMPLES X 



193 



20. Prove 



-z)<r(y-w) 



21. Prove 



22. Shew that 



23. Shew that 






24. If M + v + ?^ = 0, prove 

{ f () + f (^ + f ( 

25. Shew that 

2^(2?*) = f (*) + f ( - 

26. Shew that 



= P() + P 



u - o> 2 ). 



1 p(w) tf*(w) 
27. Prove 

f(w-*0-f(**-w) -(*-> 



/ \ / V 

- 2v)ar(u - v)v(u - w) 



M.F. 



[CH. XI 



CHAPTER XL 
JACOBIAN ELLIPTIC FUNCTIONS. 

77. The Values of <p(w) when o^ is Real and o> 2 is Purely 
Imaginary. Let (0^ = 0,^ ft) 2 = iQ 2 , where Q x and Q 2 are real and 
positive ; then 



i=0 m = -< 

The two terms in this bracket are conjugate complex 
numbers, so that g 2 is real. Similarly it can be shewn that 
#3 is real, and that, if w is real, $>(w) and <(p'(w) are real ; 
while if w is purely imaginary, $(w) is real and $?(w) is purely 
imaginary. 

Thus e 1 = p(Q x ) and e 3 = $>(il z ) are real ; also, since e% = e l e B , 
e 2 = ^(Q 1 + ^Q 2 ) is real. Hence the three roots of the equation 
4# 3 g 2 x g s = are all real. 

Now consider the values of $(w) at points on the rectangle 
OABC (Fig. 74), where A, B, C are the points Qj, S^ + iQ^ ^Q 2 , 
respectively. 

v/k 



FIG. 74. 



(i) If w u is real, small, and positive, %>'(u) is, large and 
negative ; also, when u = l l , $>'(u) vanishes. Between these points 
on the real axis %>'(u) is continuous, and has no zero values 
( 72, Th. V.). Accordingly, between and Q lt $>'(u) is negative ; 
so that $>(u) decreases continuously from +oc to e r 



77] VARIATION OF p(w) 195 



Now ^ /2 (^) 

Therefore since, as u increases from to Q x , ^(u) decreases 
continuously from +x to and $>(u) decreases continuously 
from +x> to e lt e l is the greatest root of ^x s g^cg s = 0. 

Again, p'(u) = <J{4<$ B (u) g$(u) g s } ; but between and 
Q 1? ^X^) is negative and 4{p(u) j}{|i(it) ^}{p(u) e 8 } is 
positive. Therefore 



Hence, if x <^(u), 

r dx 

U=\ , 3 

J % v/ "j T?i/ " \j rt! 

provided a;^^. In particular, 

c?o? 



fee 
e, 



(ii) Let w = iv, where v is real ; then 
' 



where 0^(^) = 40 3 (t;) 

As in (i) it can be shewn that e 1 = 0(Q 2 ) is the greatest root 
of 4x^g 2 x-\-g 3 = ) and that 

dx 



f* 

= 

J ^i 



Thus j_ or e 3 is the least root of 4# 3 # 2 # ^ 3 = 0, and 



Also, as v increases from to Q 2 , 0(v) decreases from -f x to 
e p so that ^(w ) increases continuously from oo to e 3 . 

Since ^ + 62+63 = 0, and e l '^>e f> '^>e 3 , it follows that e l must be 
positive and e s negative. 

(iii) Let w = u + il. 2 , where u is real ; the% since 



196 FUNCTIONS OF A COMPLEX VARIABLE [CH. xi 



and p'(u + i& 2 ) are rea l- As u varies from to Q I} 
increases from e s to e 2> and p'(u+iQ 2 ) is positive. 
Therefore, between and f2 1? 



/7r 
so that Q 1= '" 



(iv) Let w = Qj + w, where v is real ; then, since 



iv) is real and ^'(Q^iv) is purely imaginary. As v 
varies from to Q 2 , ^(Q^w) decreases from 6 X to e 2 . Thus, 
if 0(v)= ^(Qj + iv), between and H 2 4>( v ) varies from e x 
to e 2 and 0'C^) ^ s positive. Therefore, since 



da; 
Hence 



Accordingly, as i; passes round the rectangle OABC, p(io) 
decreases continuously through all real values as follows : from 
+ 00 at O to e 1 at A ; from e 1 at A to e 2 at B ; from e<> at B to e 3 
at C ; and from e 3 at C to oo at 0. 

Let p be any real quantity, and let f be the point on the 
rectangle for which p(f)=JP- Then, since $>( )=p and <p(w) 
is of order 2, every point w such that #>(tt;)=^) must be con- 
gruent to f or Therefore, for every point within OABC, 
p(w) is imaginary or complex. 

Example. Shew that 

(i) 
(ii) 



78. Geometric Application.* Consider the curve given by 

* Cf. Appell et Lacour, Fonctions Elliptiques, 68-63. 



77, 78] GEOMETRIC APPLICATION 197 

To each value of x correspond two non-congruent values w 
of the argument. But #>'( w)= ^'(w); hence to each point 
(x, y) on the curve there corresponds only one non- congruent 
value of w, and the curve is symmetrical about the aj-axis. 

Condition that three points should be 'collinear. Let M I} M 2 , 
M 3 , be the three points in which the line y mx c = Q cuts the 
curve. The corresponding values w l , w 2 , W B , of w are zeros of 

$>'(w) m$(w) c. 

Now the only pole of this function is at the origin, and is 
of order 3 ; thus 



Wi + ,,+,, 

where X and //, are integers. 

This relation is necessary, and it is also sufficient. For, if 
w l + w 2 +w z = 2\co l + 2yuft> 2 , let the line M^Mg cut the curve 
again in the point M' of argument w'\ then 
w l + w z + w' = 2X'o> 1 + 2yu'ft> 2 . 

Hence w 3 and w' are congruent, so that W coincides with M 3 . 

Tangents. If the tangent at 'w^ meets the curve again at 
' w ' w+ 2w 1 = 

Thus w l = w/2 + Xo)! 

Accordingly, from any point ' w ' four tangents can be drawn 
to meet the curve in the four points whose arguments are 



Points of Inflection. At a point of inflection 

3it; = 2Xft) 1 + 2^(o 2 ; 
so that ^ = 2Xa> 1 + 2 M a> 2 

o 

Thus there are nine points of inflection with arguments 

2^ 2a> 2 4fa? f 4o) 2 2 
~~ ~ 



~3~' ~3~' 3 ' ~3~' 7 :> ' 3 ' ~~~3 ' ~ ~~3~ 
and they lie three by three on straight lines. 

Case in which ca l is real and w z is purely imaginary. Let 
then 



2 



where gj, e 2 , e 3 , are real, and ej 



198 



FUNCTIONS OF A COMPLEX VARIABLE [OH. xi 



As w varies along OA (Fig. 74) from to Q lt the point (x, y) 
passes up the right-hand branch of the curve of Fig. 75 from 




PIG. 75. 

y= o to A(e 1? 0). For values of w between Qj and 
y is imaginary. As w varies from Q 1 + iti 2 to iQ 2 , (x, y) passes 
from B(e 2 , 0) round BCD to D(c 3 , 0). For values of W between 
iQ 2 and 0, y is imaginary. The corresponding negative values 
of w give the other two arcs. 

There are only three real points of inflection, 0, 2Q x /3, and 
2QJ3, the first being at infinity : they are collinear. 

Example 1. Shew that the necessary and sufficient condition that the 
six points whose arguments are w lt w 2t ...w 6 , should lie on a conic is 



Example 2. Shew that the necessary and sufficient condition that the 3n 
points w lt W<L, ... w sn , should lie on a curve of degree n is 



Consider the three 



79. The Jacobian Elliptic Functions, 
functions: 



78, 79] TRANSITION TO JACOBIAN FUNCTIONS 199 



They satisfy the equations : 

ft) 2 ) = - <h( 
ft> 2 ) = 

2ft),) = - 3 (u), ^ 8 (w + 2o> 2 ) = <fa(u). 
Again, by formula (A) of 76, 



Thus the two values of x/{^(X)~~ e i}> *J{p( u )~~ e z}t 
are the uniform functions <j>i(u), $> 2 (u), ty^u), respectively. 
If those values of the three functions are taken which are large 
and positive when u is small and positive, 



Now p / (^) = 2 

Also, it is easy to shew that 

f'(u) = - 20 1 (w)0 2 (w)0 8 (w). (Cf. 76, Example 1.) 

Hence ^iX^) = ^( u }^( u )- 

Similarly <j> 2 '(u)= -^(u)^^), ^ 3 / (^)= - 0iO)02 ( u )- 

Next, let to! be purely real and co 2 purely imaginary, and denote 
them by Q x and iQ 2 respectively ; then, since p(u) ^ e l >- 6 2 > 6 3 , 
provided < it ^ Qj, 

- ViC^i) = 0, 
Similarly 



Accordingly, if 



these three functions will satisfy the equations : 
^)= -S(tt), S 



200 FUNCTIONS OF A COMPLEX VARIABLE [CH. xi 



S(0) = 0, C(0) = l, D(0)=l; 

^ = 1, 0(00 = 0, 



Also S(, C(u), D(u), have simple poles at u = il 2 \ and 
is odd, while C(^), D(u), are even. 

Thus S(u), C(u), D(u), are elliptic functions of periods 
2i0 2 ; 4Q lf 2Q 1 + 2iQ 2 ; 20! , 4i0 2 ; respectively. 

Now let S(u) = sn(v), 
where ^ 

and let 



so that & and /<;' are positive proper fractions such that k 2 + A/ 2 = 1 . 
Then sn(i>), cn(v), dn(?;), satisfy the equations : 

sn (v + 2K) = - sn v, sn (v -f '2iK') = sn (v) ; 
en (v + 2K) = - en v, en O + 2^K') = - en (v) ; 
dn (v + 2K) = dn v, dn (v + 2iK x ) = - dn (v) ; 
sn' (v) = en (v) dn (v) ; en' (v) = sn (v) dn (v) ; 

dn'(-u) = - 7c 2 sn (v) en <V) ; 
sn 2 (v) + en 2 (v ) = 1 ; k 2 sn 2 (v) + dn 2 (v) = 1 
sn(0) = 0, cn(0) = l, dn(0) = 
sn(K) = l, cn(K) = (), 



/ 



Also sn(v), cn(V), dn(v), have simple poles at v = iK'; and 
sn (v) is odd, while en (v) and dn (v) are even. 

Again, since <f> 3 (u) or <J(^(u) e s ) decreases continuously from 
to V( e i"~ 6 3) as u increases from to O 15 sn.(v) increases 



79] THE JACOBIAN ELLIPTIC FUNCTIONS 201 

continuously from to 1 as v increases from to K ; accordingly, 
if z = sn(v), 



where the positive value of the radical is taken between 
and 2=1. Hence 

dz 



and therefore sn (v) is the inverse function of 66. In particular, 
K 



- 



so that K is identical with the K defined there. 
Moreover, since 

*--mz*s=&=' 

r dx 



(77) 



Q 2 can be obtained from Q x by replacing e lt e 2 , e s , by e s , e 2 , 

I / Q g 

e lt respectively. Thus K' is the same function of /v/( J 

or k' that K is of A/( J ) or &; so that K' is identical with 
the K 7 of 66. Ve i-V 

These three functions sn(^), cn(f), dn(t;), are the Jacobian 
Elliptic Functions ; their periods are : 4K, 2iK'; 4K, 2K + 2iK'; 
2K, 4iK'; respectively. 

Since ^(^) e 3 = ^(u), 

/? m ._, /? 

/ \ ^1 ^Q 

(f\(f)j\ /> * ' 

e3 -sn 2 (.a)' 

//g g \ 

where v = ,J(e l e^).u and ^ = /y/( 2 3 j- This equation gives 

the relation between the Jacobian and the Weierstrassian elliptic 
functions. 



Example. Invert the function 

rdx 
v 



, \ H). 



202 FUNCTIONS OF A COMPLEX VARIABLE [OH . xi 

Poles of sn(v), cn(v), dn(v). From the equation 

it follows that 



w^ = <f 2 = - i, ( 75) 




CQ)_ fa(u) D(u) 
so that Lim ;. r ^ 2 -^ = e ~ w^^ 

and 



Accordingly, if I is the residue of sn(t> ) at tQ 2 , the residues of 
cn(t;) and dn(v) at this point are II and ikl. 

The function su(v) has poles at iK' and 2K-f iK r , at which the 
residues are I and I respectively; it is therefore of order 2. 
Similarly cn(t>) and dn(v) are both of order 2. 

Note. The two periods K and iK' are not, like o^ and o> 2 
in the case of the Weierstrassian functions, independent of each 
other : they are connected by the relations 

T 7-_ 1 dx ' 

~ 
where 



Example. Prove 

(i) E== ( K dn 2 (w, lc)du ; (ii) E'= \*'&u?(u, k')du. 

Jo Jo 

80. The Addition Theorems. Consider the functions of u, 
\(u + v) and cn(u)cn(u + v) cn(V): they both have 
periods 2K and 2iK', and simple poles at iK' and v + iK'. 
Hence they are of order 2, with simple zeros at u = and 
u = ' v ; so that 



where C is a constant. 



79, 80] THE ADDITION THEOREMS 203 

When u is. small, 

...){cn( usn(v)dii(v) +...} cn(t>) 

7 r ; - 

usu(v)-\- ... 



Now let 16 = 0; then C= dn(v); so that 

cn(V) cn(t6+ 1>) + sn (u) dn(v) sn(ic + v) cn(-y) = 0. 

If in this equation u and -y are interchanged, it becomes 
cn(t>) cn(u + v) -f sn(v) dn(u,) sn(i6 + 1?) cn(u) = 0. 

Hence, solving these two equations for sn(w + v), and writing 
c 1? cZj, s 2 , c 2 , cZ 2 , in place of sn(u), cn(u), dn(u), sn(^), cn(v), 
v), respectively, we have 



. 
1 d l s l c. 2 c/ 2 



Similarly 



In like manner, by considering the functions sn(u)sn(u-\-v) 
and dn(u)dn(y + i>) dn(v), it can be shewn that 

_ cZj d. 2 k 2 s 1 s 2 Cj <? 2 



COROLLARY. If in these formulae v is written for v, they 
become 



Example. Prove 



204 FUNCTIONS OF A COMPLEX VARIABLE [CH. xi 

Duplication Formulae. In the addition formulae make v = u; 
then, if sn(u), cn(u), dn(u), be written s, c, d, respectively, 
/tt . 2scd 

SEE sn(2tt) = = _, 

J. K S 



From these formulae the following can be derived : 

^FTTC = F " D-C ' 



D + C 7c /2 1-D 



7/2 l-C 

r+c = ^D^c 

Example. Shew that 



From the addition formulae it follows, since 
sn(K)=l, cn(K) = 0, 



that 



, , - 

dn(^) . dn(u) dn(u) 

Hence, if u tends to iK', 

su(K + iK') = jj cn(K + 'iK / )= -^, dn(K+iK / )-0. 
Now in the addition formulae put v = K + iK' ; then 



By repeated applications of these formulae the following can 
be derived : 

= cn(V), 



K x )= -dn(u); 



80, 81] JACOBI'S IMAGINARY TRANSFORMATION 205 

2iK / )= cn(u), 



Sample. Prove 

Again, in the addition formulae let i> tend to iK' ; then 



sn(u) 

Thus the residues of sn(u), cn(w,), dn(i(,), at iK' are I/A;, i/k, 
i, respectively. 



81. Jacobi's Imaginary Transformation. Letx = sn(iu,k')', 
then 



Now put x = iy/J(I 2/ 2 ) ; then 



so that y = sn(u, /c). 

Thus sn(iu,k')=i su(H ' J ;\ 

cn(it, A;) 

To determine the sign let u tend to zero ; then, since 

T . sn(m, k') 
Lim . V ? 7 : = 1, 
n ^o t sn(u, /c) 

the + sign must be taken ; so that 



1 . 
, k) 

Again, cn(m, k') = J{l sn' 2 (m, &')} = - 



To determine the sign let u ; thus 

cn(m, &') = -, j-. 
cn(u, k) 



Similarly du(iu, Jc) = / > ? . 

GU(U, k) 

Example. Shew that 



sn 2 (m, k') sn 2 ^, k) 



206 FUNCTIONS OF A COMPLEX VARIABLE [OH. 

EXAMPLES XI. 

1. Prove / . 

Jo 



2. Shew that, if sn 11= sin <, 

rdu sn u dn u 
iTcn^ 

3. Prove the following identities, in which D denotes 1 -J 

(i) sn (u + v) sn (u -v} = (c 2 2 - Cl 2 )/D = ( Sj 2 - s 2 2 )/D : 

(ii) {Icn(u + v)UIcn(u-v)} = ( Cl c 2 yiT>', 

(iii) {ldn(u + v)}{ldn(u-v)} = (d l d. 2 ) 2 ID; 

(i v) sn (u v) en (u qp v) = (s^d^ s 2 c 2 c 1 )/D ; 

(v) sn (u v) dn (u T v) = (s^c^ s^c^/D ; 

( vi) 1 + en (u + v)cu(u-v)=(cS + c 2 2 )/D 5 

( vii) sn (u + v) en (u -v) + tm(u- v) en (u + v) = 



4 Prove \ 2 / V 2 



en ?6 + en v 
en 



2 
5. Shew that, with the notation of Example 3, 



D 3 
-v}an(u-v) an"(u-v) 

6. Verify the identity 



where S = sn (u + v) sn (u - v) sn (u + w) sn (u - w), 

C = en (u + v) en (u v) en {u + w) en (w w), 
D = dn (w + v) dn (u -v)du(u + w)dn(u- iv). 

7. If S = sn u sn (u + K), verify, that : 



(ii) 
(iii) 

( 1 _ v 

Deduce that 



8. Shew that the function of M, 

snu cnu dn w(sn 2 y sn 2 2<;) + snv cnv dn v(sn 2 ^ sn 2 ^) 

+ sn -^ en ?<; dn w(sn 2 ^ - sn 2 v), 
has periods 2K and 2i'K' ; and prove that u = iK f - v-w is a simple zero. 

9. Prove that the function of u, 

sn 4 w(sn 2 v - sn 2 w) + sn 4 v(sn 2 w - sn 2 w) + sn 4 'w(sn 2 *i - sn 2 #), 
has periods 2K and 2^K' ; and shew that it has four simple non-congruent 
zeros, i\ w. 



xi] EXAMPLES XI 207 

10. Verify that 



11. Prove l-dn(2n) gV 



12. Prove dn(w, k) = 'sn(K' - iK - iu, k). 

13. Let w=sn 2 (z, k\ and let A, B, C, be the points K, K + ^K', ^K', respec- 
tively, in the 2-plane. Shew that, as z passes round the rectangle OOAB, 
w passes through all real values from - cc to 4- oc . If COAB is a square, 
what is the value of k ? Ans. 

14. The coordinates of two points are connected by the equation 

~ 



Shew that, as (#, y) describes the boundary of the rectangle COAB of the 
previous example, (X, Y) describes the complete boundary of a quadrant 
of a circle of unit radius. 

15 Prove 



dn ( u + v) dn (u - v) k'~ 



cn(u + v) cn(u-v) afi 
17 Shew that (i) Lim Bn " u 


?=* 


,..., T . udnu snu 1 2- 
nil) l/im 





18. Establish the expansions 



en 2*= 1 - 



where \u\ < K'. 

19. If the tangents from the point P on the cubic x=.yp(w\ y = p(w\ meet 
the curve in A, B, C, D, shew that the pairs of lines : AB, CD ; AC, BD ; 
AD, BC ; intersect at points Q, E, S, on the curve ; also shew that the 
tangents at P, Q, K, S, intersect at a point on the curve. 



[CH. xn 



CHAPTER XII. 

LINEAR DIFFERENTIAL EQUATIONS. 
82. Continuation of a Function by Successive Elements. 

03 

Let P(z, a) denote a Taylor Series ^c n (z a) n with circle of 

o 
convergence C ; then, if z 1 is any point within C, this function 

CO 

can be expanded at z l in a Taylor Series ^jCn(z z^) n , which we 



o 



denote by P-^2, z^). The circle of convergence C x of this series 
will either touch C internally or lie partly outside C : in the 
latter case P^z, zj gives the continuation of P(z, a) in the part 
of Cj outside C (55, Th. II. Cor.). The two expressions P(z, a) 
and P^z, Zj) are called Elements of the function. The radius of 
C x will be the distance from z l to the nearest singularity of the 
function ; so that, if C x touches C internally, the point of contact 
must be a singular point. 

It may happen that no part of the circumference of C can be 
found, however small, which does not contain singularities of the 
function : in this case the function cannot be continued beyond C. 
If, on the other hand, the function can be continued beyond C, 
the process can be repeated with each new domain so attained. 
The aggregate of the elements thus obtained defines an Analytic 
Function. 

Note 1. If the only singularity of the function is at infinity, 
the original element gives the complete function. 

Note 2. If f(z) is holomorphic at infinity, the corresponding 
element is obtained by continuing /(1/f) to a domain of centre 
f-OL 

A particular point b can usually be approached by different 
continuations from a; and it is possible that the function may 
thus attain different values at 6. If the values are always the 



82, 83] HOMOGENEOUS LINEAR EQUATIONS 209 

same, the function is uniform. The case of multiform functions 
requires more particular investigation. 

Join a and b by a path L: on L take points a, z v z 2 , z 3 , ... , 
such that each point lies in the domain of the preceding one. 
Then the corresponding elements give a value of the function 
at each point on L. If no singularity lies on L, the points 
z i> z z> z s> " > can ^ e chosen so that, after a finite number of steps, 
a domain is reached which contains 6, and thus a value of the 
function at b is obtained. 

This value is independent of the set of points z lt z z , z s , ... , 
selected. For, let a set of points z ni , z n _ 2 , ... , z Ur , be interpolated 
on that arc of L which joins z n and z n+l ; then, if the elements 
corresponding to these points are employed in the process of 
continuation, the same value is attained at z n+l , since the arc lies 
entirely in the domain of z n ( 55, Th. II. Cor.). Now, any two 
sets of dividing points, z lt z 2 , z 3> ... , and z^, z 2 ', z s ', ..., can 
be combined, and other points, if necessary, interpolated between 
them, in order that each point of the new set may lie in the 
domain of the preceding one. Hence it follows that each of the 
original sets gives rise to the same functional value at b. Thus, 
if the function varies along a line which does not pass through 
a singularity, the set of values obtained at points on the line 
is always the same. 

Again, since the points z v z 2 , z 3 , ... , can be chosen so that each 
not only lies in the domain of the preceding point, but also in the 
domain of the succeeding point, it follows that, if the value at b 
be taken as initial value, and if the path L be retraced from b to 
a, the same set of values will be obtained at all points of the line. 

Finally, if any two paths L and L' are drawn from a to 6, 
such that no singularity lies between them, they will lead to the 
same value at b ; for otherwise the closed contour made up of L 
taken from b to a, and of L' taken from a to b, would enclose at 
least one branch-point of the function, which contradicts our 
hypothesis. 

83. Homogeneous Linear Differential Equations. A linear 
differential equation 

d n w .d n ~ l w .d ll -*w 



210 



FUNCTIONS OF A COMPLEX VARIABLE [CH xn 



which involves no terms independent of w t is said to be Homo- 
geneous. We shall assume that the coefficients are uniform 
functions with no singularities except poles in the region con- 
sidered. A point which is an ordinary point for all the coefficients 
is called an ordinary point of the differential equation, while a 
point which is a singularity of any one of the coefficients is 
called a singularity of the equation. If f is an ordinary point, 
and if a is the singularity of the equation nearest to the 
interior of the circle z | = | a f | is called the domain of 
If the equation is of the first order, its solution is 



where C is an arbitrary constant. Accordingly, it is only 
necessary to consider equations of order higher than the first, 
We shall, indeed, confine our attention to equations of the second 
order; but the methods employed can be applied, with suitable 
modifications, to equations of higher order. 

THEOREM. In the domain of an ordinary point f the differ- 
ential equation c L^p(^ z \ ( - + q(z}w (A) 

possesses a unique integral w(z), which is a holomorphic function, 
and which, with its first derivative, acquires arbitrarily assigned 
values (the initial values) when z = f, 

Let Mj and M 2 be greater than or equal to the greatest values 
of |_p(z)| and q(z) on the circle z f| = R, where R<|a fl 
and a is the nearest singularity of the equation to f. Then 
(35, Cor. 1) the functions 

M 



*-jpb 



^(z) = 



R 



satisfy the inequalities 






where n = 0, 1, 2, ____ The functions <p(z) and \fr(z) are called 
Dominant Functions, and the equation 

z) ^ 



is called the Dominant Equation. 



83] EXISTENCE OF AN INTEGRAL 211 

Now, if a function w(z) is holomorphic in the domain of f, it 
can be expressed in that region in the form of a convergent series 

c + c 1 (z-) + c t (z-&+..., (I) 

where < = !, *^>, (=0, 1, 2, ...). 



Tl : 



But if this function w(z) is an integral of equation (A), and 
if arbitrary values have been assigned to w(g) and w'( ), the 
corresponding value of w"(f ) can be obtained by substituting 
for 2 in the equation. Likewise, if the equation is differentiated 
repeatedly, and f substituted for z, equations 



are obtained for n = %, 4, 5, ... ; thus the coefficients c , c lt c 2 , ..., 
can be found. 

Similarly, if W(z) is a solution of equation (c), holomorphic 
within | z f | = R, 

where c n ' = , /&! 

and 



Now let I iv (g ) | and | to'(f ) I be assigned as initial values to 
and W(f) J tnen ' ^ rom equations (A), (B), (c), (D), and (E), it 
follows that, for all values of n, W (n) (f ) is real and positive, and 



Accordingly, if the series (n) can be proved to be convergent, 
the series (l) will also be convergent, and w(z) will be holo- 
morphic in the domain of f. 

Let 2-f=RZ; then 

. , (m) 



212 FUNCTIONS OF A COMPLEX VARIABLE [OH. xn 

where c n " = R n c n ' ; and equation (c) becomes 



so that l- 



In this equation put Z = ; then, since 





we have 

Now let M x be chosen so great that RM X > 2 ; then 
so that c'nlc" n+l <^\.. 

H - T> TIT "D^lXf 

T) , C n 9 n-t-KtM-, . JtvlYl* 

But 

therefore 

Thus series (in) converges if | Z | < 1 ; hence series (n), and 
consequently series (i), converges if | z f | < R. 

Now, if z is any point in the domain of f, R can be chosen so 
that |z f|<R< a, |- Accordingly an integral w(z) exists, 
which is holomorphic in the domain of f, and is such that 
arbitrary values can be assigned to w(g) and w'(g). 

COROLLARY 1. The integral is unique. For, if any particular 
values are assigned to w(f) and w\g), only one set of values for 
<> C 3> C 4> ' > can ^ e deduced from equations (A) and (D). 

COROLLARY 2. The integral is of the form CQW I (Z)-\-C^JO^(Z\ 
where c , c x , are arbitrary constants, and w^z), w z (z), are 
integrals of the equation. For, by means of equations (A) and 
(D), all the constants c 2 , c 3 , c 4 , ... , can be expressed linearly in 
terms of c and c r Also, by making c and c^ zero in turn, we 
see that w^z) and w z (z) are integrals of the equation. 

Integrals at Infinity. To determine whether infinity is an 
ordinary point of the equation, the transformation z = l/ is 
employed. The equation then becomes 

a(MF\ 

w, 



83, 84] SOLUTION BY INFINITE SERIES 213 

so that it is necessary that p(z) + 2/z and q(z) should have zeros 
of orders 2 and 4 respectively at infinity. If this condition is 
fulfilled, holomorphic integrals w() or w(l/z) can be found. 

A nalytical Continuation of the Integral. Let f ' be any point 
in the domain of f, and let P(z, f ) be the element of the integral 
w(z) corresponding to the domain of f '. Then, since the function 

Jp( Z , n-j(*)^p(- n-?(*)p<*. r> 

vanishes at all points common to the domains of f and f, it 
vanishes at all points of the domain of f (55, Th. III.); thus 
P(X f) satisfies the differential equation, and has the initial 
values w(') and w'(') at f. Similarly it can be shewn that 
every element obtained from w(z) by analytical continuation 
satisfies the equation. 

84. Solution by Infinite Series. An integral w(z) can be 
obtained by assigning values to c and c^ and then finding 
C 2> C 3> c t>'-> by means of equations (A) and (D) (83). In 
practice, however, it is usually simpler to proceed as follows : 

(i) if an integral in the domain of z = is required, substitute 
the series 



for w in equation (A), and equate the coefficients of powers of z ; 
a series of equations is thus obtained which enables us to deter- 
mine c 2 , c 3 , c 4 , ... , in terms of c and c l ; 

(ii) if an integral in the domain of any point a is required, 
apply the transformation z = a + to the equation, and use 
method (i) ; 

(iii) an integral in the domain of infinity can be obtained by 
applying the transformation z = l/f and using method (i); it is, 

00 

however, simpler to substitute the series ^c n /z n in the equation 
and equate coefficients. 

Note. The theorem proved in the previous section, and the 
method of solution just given, apply also to equations of higher 
order than the second. 

Legendre's Equation. Consider the equation 



-z 2 ) and q(z)= -7i(n+l)/(l-z 2 ); thus z = 



214 FUNCTIONS OF A COMPLEX VARIABLE [CH. xn 

is an ordinary point of the equation, its domain being the 
interior of the circle z \ 1. 

Let <iv = c + c l z + c 2 z 2 +... 

be substituted in the equation ; then 



so that 2.1. 

3.2.03-2^ + 71(71 + 1)^ = 
^+2)^+1)^+2-^-1)^-2^ + 71(71 + 1)^ = 0, (,, = 2, 3, 4, ..,). 

Hence . 2 =- 

~ 



2.3 1J 

> 



Therefore w = c Q w l + c^v 2 , where 

w =F(-- n+l - z 2 ] w =* 

\ 2 *w / 

If 7i is an even positive integer, the first, and if n is an odd 
positive integer, the second of these series contains only a finite 
number of terms ; so that, if n is a positive integer, one integral 
is a polynomial. 

Now, if n is even, 









2' 2 '2' (!) 2(2-l) 

n(n-l)(-2)(-8) ^ 
1 " " 



'(z); (54, Cor.) 

while, if n is odd, 



m *i _l_ 9 Q \ \9 

sF( - , + , -, ^ 



Thus, if n is a positive integer, one integral is the Legendre 
Polynomial ~P n (z\ which is also known as Legendre s Function 
of ike First Kind. 



84, 851 LEGENDRE'S EQUATION 215 

Example. Find integrals for 



Ans. Wl = l- + 6 - 



'2.5 7 2.5.8 

I ,-fl __ 

7! 10! 



85. Fundamental System of Integrals. 

THEOREM I. The integral w(z) of equation (A), 83, cannot 
have a zero of the second order at any ordinary point of the 
'equation, unless it vanishes identically. 

For if it has a zero of the second order at the point z, w(z) = 
and w'(z) = Q', hence the equation gives w"(z) = Q. Similarly, if 
the equation is differentiated repeatedly, it follows that 



so that the integral is identically zero. 

THEOREM II. If w (z), w 2 (z), w 3 (z), are integrals of the differ- 
ential equation holomorphic in the domain of f, a relation of 
the form c^z) + c 2 w z (z) + c 3 w s (z) = 

exists, where c lt c 2 , c 3 are constants not all zero. 

For if c lt c. 2 , c 3 , be chosen to satisfy the two equations 

w 3 (f ) = 0, 
(f) = 0, 

the integral w(z) = c l iu l (z) + c 2 w. 2 (z) + c^v 3 (z) and its first deri- 
vative vanish when z = . Hence, by Theorem I., w(z) is 
identically zero ; so that 

c l w 1 (z) + c z w 2 (z) + c s w 3 (z) = 0. 



DEFINITIONS. Two integrals are said to be linearly inde- 
pendent if their quotient is not a constant. Two linearly 
independent integrals are said to form a Fundamental System 
of Integrals. Such a system can always be obtained by making 



From Theorem II. it follows that if the integrals w (z) and 
w 2 (z) form a fundamental system, any integral can be expressed 
in the form c l w l (z)-i-c 2 w 2 (z) t where c and c 2 are constants. 



216 FUNCTIONS OF A COMPLEX VARIABLE [CH. xn 

Again, if c l w 1 (z) + c 2 w 2 (z) = Q, then A(2) = 0, where 

A(z) = 



Conversely, if A(z) = 0, a relation C 1 w 1 
for, if A(3) = 0, 

w i( z ) _ w z( z ) 



= exists: 



and the integral of this equation gives a relation of the type 
required. 

THEOREM III. If the integrals w^z) and w 2 (z) form a funda- 
mental system in the domain of f, A(z) cannot vanish in that 
domain. 

For let W 1 (^), W 2 (z), be another fundamental system ; then 

so that ' Jl ' (z) ' Wl(z) 



where 



'12 > 



The determinant D cannot vanish, since W^z) and W 2 (0) are 
linearly independent. But W-^z) and W 2 (z) can always be 
chosen so that, at any assigned point z in the region, 

W 1 () = l, W^) = 0, W 2 (2;) = 0, W 8 '(*) = l. 
Hence A(z) is non-zero at every point of the region. 

THEOREM IV. If two linearly independent functions w^z) 
and w 2 (z) are holomorphic in the neighbourhood of f, and are 
such that A(f ) =^= 0, a homogeneous linear differential equation of 
the second order can be constructed, of which they are integrals, 
and of which f is an ordinary point. 

For if the functions p(z) and q(z) are defined by the two 
equations w ^ -p(z) Wl '(z) - q(z)w 1 (z) = 0, 



then 
where 



p(z) = \(z)/A(z), q(z) = - A 2 (^)/A (z), 
^'(z\ w 2 (z) , 2 Z 



85] DETERMINANT OF A FUNDAMENTAL SYSTEM 217 

Now the numerators and denominators of these two fractions 
are holomorphic, and A(f)^0; hence p(z) and q(z) are holo- 
morphic near z = g. Accordingly w^z) and iu 2 (z) are integrals 
of the equation w" =p(z)w' + q(z)w, 

of which f is an ordinary point. 

Example. Find an equation which is satisfied by 

w -I ^.Lif! 1 . 3 . 2 6 1 . 3 . 5 . 2 s 

Ans. w"= zw' + w. 

EXAMPLES XII. 

1. Find integrals w lt w 2 , for w" + a 2 tv = Q, such that, when 2=0, ^ = 1, 
Wi = 0, w 2 = 0, w 2 ' = 1 . A ns. w l = cos az, w 2 = a~ l . sin az. 

Find integrals in the domain of 2 = for equations 2-10. 

2 3 4 2 4.7 2 9 



3. (2 2 - F) /' + zw' -w=0. Ans. w l = z, w 2 = V(^ 2 - 2 2 ). 

4. (l+2 + 2 2 )w" + 2(l + 22)?^' + 2w = 0. Ans. ie 1 = ^ 5 , w a = -g. 

5. (2-l)(2- 

6. 

7. (l- 



8. (l-2 2 )w"-( 

W =F +M) _ 



9. (1 - 2 s ) M' 

^7M. ^=^,t^ = F(-, -f, , 2^), ^ 3 

10. ^"-2 2 z^" + 22?^-2w=0. 

22 s 22 



11. Find integrals in the domain of 2=1 for 2(2-2)%'" 

A ns. w 1 =2(2 - 2), w 2 = 2(2 - 1) + 2(2 - 2) log {2/(2 - 2)}. 

12. Find integrals in the domain of 2= - 1 for w" - (1 +z)w' -w=0. 



218 FUNCTIONS OF A COMPLEX VARIABLE [OH. xii 

Find integrals in the domain of infinity for equations 13-15. 

13. %" = (l-2s)3V + 2w. Ans. w^e 1 '*, w 2 =e~ 2/ *. 

14. Z 4 w" + 2z 3 iv' + a 2 iv=0. Ans. w 1 = coB(a/z) J w 2 =sin(a/z). 

15. z*(z*-l)w" + 2z(z* + I)w'-2w=0. Ans. iv^z^-l), iv 2 =z/(z 2 -!). 

16. Find an equation which is satisfied by 



Ans. 

17. Find an equation which is satisfied by iv 1 =z, w^e 1 . 

Ans. (z-l) w" - ziv + iv = 0. 

18. Shew that, if n is a positive integer, the equation 



has integrals P(^) and P(^)log ( i!^+Q(z), where P(0) and Q(z) are 
polynomials of degrees n + 1 and n respectively. 



CH. xiii, 86] 



CHAPTER XIII. 

REGULAR INTEGRALS OF LINEAR DIFFERENTIAL 
EQUATIONS. 

86. Integrals in the Neighbourhood of a Singularity. 

Consider a homogeneous linear differential equation of the 
second order, of which f is a singularity and of which w lt w z , 
form a fundamental system of integrals at z. Let z describe 
a closed circuit which encloses f but no other singularity of 
the equation, and let w l and w z be the analytical continua- 
tions of w l and w z obtained when the variable has completed the 
circuit. These two integrals ~w lt w z , form a fundamental system ; 
for, if not, a relation c 1 i(; 1 + c z w z = would exist. Consequently 
the function cjw l + c z w z would vanish at all points to which it 
can be continued (55, Th. III.); and therefore, retracing the 
circuit, we would obtain the relation c^w^c^w^O, which con- 
tradicts our hypothesis. Accordingly 

w^ = c ll w l + c lz w z , w 2 = c z] w l 
where D= 



_Now let W = Xw 1 + /*U' 2 , and choose the constants X, yu, so that 
W, the value attained by W after the description of the closed 
circuit, satisfies the equation W = pW, where p is a constant ; then 



Therefore, since w lt w z , form a fundamental system, 
c n -p, c 21 



so that 

''12 



= 0. 



220 



FUNCTIONS OF A COMPLEX VARIABLE [OH. xm 



This is known as the Fundamental Equation belonging to the 
singularity If a root of this equation is substituted for p in 
equations (1), values of X and JUL are obtained such that W^pW. 

Neither of the p's can be zero, since D=/=0. If /o = l, the 
corresponding integral W will be uniform in the vicinity of 

THEOREM. The fundamental equation is independent of the 
original fundamental system selected. 

Let W x , W 2 , be any other fundamental system, and let 



so that the new fundamental equation is 



'21 



6 12 , 



0. 



&22-P 

N ow, if W 1 = ai l w l + a 12 w 2 , W 2 = a_ 
then W x = a n w t + a 12 w 2 , W 2 = a^^ + a 22 w 2 . 

= a n( c n w i + c iz 
Accordingly b n a u + 6 12 a 21 = anCj! + a 12 c 21 , 



Similarly 
Therefore 



(2) 



21 a n + 6 22 a 21 = a a c u 



'12 



a. 



21 



^12' 



1 > 



'21 



Hence 



600 p 



'2-2 



C u /O, C, 



21 



C 22~~P 



Fundamental System associated with the Fundamental Equation. 
There are two cases to consider : (I.) when the roots of the 
fundamental equation are distinct, and (II.) when they are equal. 



86] FUNDAMENTAL EQUATION 221 

I. Let the roots p lt p. 2 , be distinct ; then there are two integrals 
W lt W 2 , such that 



Now let r 

Then, if O i = ( z -^r l) O z = (z-ft\ 

1 = p^O-L , 9% = p 2 9. 2 ; 
so that W^ and W 2 /0 2 are uniform functions in the vicinity of f . 

Accordingly W, = (z - f)^W, W 2 = (z-fl^ 2 (3), 
where ^iC 21 ) an( i ^C 2 ) are uniform in the vicinity of 

The integrals Wj_ and W 2 are linearly ^ndependent. For, if 
not, an equation c x W 1 + c 2 W 2 = 0, and consequently an equation 
c i/iW 1 4-c 2 /o 2 W 2 = would exist. But these equations can only 
exist simultaneously if p 1 = p 2 , which contradicts our hypothesis. 

II. Let the roots be equal; then (c n c 22 ) 2 -f 4c 12 c 21 = 0. 

We distinguish between the cases: (i) when c 12 and c 21 are 
both zero ; and (ii) when they are not both zero. 

(i) In the first case p = c n = c 22 , and w l = pw 1 , w 2 = pw z . From 
equations (2) it follows that, no matter what system is originally 
selected, these equations hold. Accordingly 

^l = (^-D r ^lW> ^ = (*-?) r ^2(3)> 

where V'l^)* ^2(2), are uniform in the vicinity of f, and 



(ii) In the second case, let W be the integral found to satisfy 
the condition W = pW, and let w be any linearly independent 
integral. Then w = c l W + c z w t and the fundamental equation 
becomes p _^ Q 

C l> C 2~ ( 

where <r is the quantity to be determined. 

Accordingly, since the roots are equal, c 2 = p ; therefore 
w = c^ W + piv. 

Now replace W by W lf where pW l = c l W, and write W 2 for 
Then W 1? W 2 , form a fundamental system such that 



222 FUNCTIONS OF A COMPLEX VARIABLE [OH. xm 

Hence W^z 

where ^(z) is uniform near f. 



W 

Again, I 



but, if = - log (2; - f ), = + 1. Therefore 



sothat 



is uniform near f, Consequently 



where ^z * s uniform near 



87. Regular Integrals. If the highest negative powers of 
(z ) in the Laurent Expansions for ifs-^z) and \^ 2 (2) are finite, 
the integrals W x and W 2 are called Regular Integrals. Now 

the quantities r = ^ . Log /o are not definite, but have values 

differing by integers. Hence, if the integrals are regular, the 
values of r : and r 2 can be chosen so that 



where a and b are non-zero. If a is the nearest singularity to 
these expansions are valid for |0 f |< | a f | . Thus 

w-'. | 

D'^i(^2^1og(2-D.J 
where r l r 2 is an integer or zero. 

For the first integral r x , and for the second integral the greater 
of the two quantities r x and r 2 , is called the Index at the point f 

It is only possible to carry out the theory completely when 
the integrals are regular ; and we shall therefore, in what follows, 
confine our attention to equations whqse integrals are regular. 

Condition that the Integrals at a Singularity should be 
Regular. If w is an integral of equation (A) of 83, a linearly 
independent integral can be found as follows. 



86, 87] REGULAR INTEGRALS 223 

Let w. 2 = wAvdz be substituted in the equation; then 



so that v = -'~ 

The integrals w l and w 2 are linearly independent. For, if not, 
therefore 



Hence, differentiating, we- have c. 2 v = 0. But v is not identically 
zero ; therefore c 2 = 0, and consequently c x = 0. 
Thus w^ w 2 , form a fundamental system. Also 

' " 
jV, 

Since every other integral can be expressed linearly in terms 
of w l and w 2 , it is only necessary to find the condition that w l 
and We, should be regular. 

Now we can always choose w l so that v is free from logarithms. 
For, if w l is free from logarithms, while w 2 contains them, 



Thus 





and therefore v = ^= = ^ r - = v. 

dz\w l 

Hence v is uniform in the vicinity of Consequently 

z) or w^v is also free from logarithms. 

Again, if w l is replaced by W 1? where cw l = pW lt then W : = pWj , 



so that V is free from logarithms. 

O 

Now write w l and v for W l and V ; then 

MJ. = /o^j , w 2 = p (^ 4- ^ 2 )- 

Thus i^j and w z can be chosen so as to have the forms of 
formulae (A). 



224 FUNCTIONS OF A COMPLEX VARIABLE [CH. xm 

In order to determine the index of A (2), three cases have to 
be considered. 

I. Let w t and w 2 be free from logarithms, and let r^r z . 
Then the index of A(z) or wfv is 2r 1 + (r 2 r 1 l) = r 1 + ?% !, 

since v = -r- (w z jw^. 

II. Let 10! and w 2 be free from logarithms, and let r 1 = r 2 . 
Then, by subtracting a multiple of w 1 from tu 2 , we can remove 
the first term of w z , and thus get Case I. 

III. Let w 2 involve a logarithm. If r z = r l} then 



where <f>(z) is holomorphic near f. Hence the index of A (z) is 
2r l l=r l + r z I. If r 2 -<r 1 , the index of v is r 2 r x 1, so 
that the index of A (z) is r 1 + r 2 1. If r 2 > r x , then, adding w 
to w 2 , we get the case r 1 = r 2 . 

Hence in every case the index of A (z) is 7^ + ^ 1. 

Now p(z) = \(z)/A(z), q(z)= - A 2 (z)/A(z) (Theorem IV. 85). 
But a circuit about multiplies A(z), A^z), A 2 (0), by the same 
constant D (86); hence ^9(2;) and q(z) are uniform in the 
neighbourhood of f. 

Again, since A(0) has the index r + r z 1, A 1 (0) or j- A(0) must 

have an index =r l -}-r z 2 J and A 2 (is) or w 1 wfv %w l /2 vw l Witf 
an index > r x + T 2 3. 

Accordingly, in order that the integrals should be regular in 
the vicinity of the singular point it is necessary that the 
equation should be of the form 

* dw O 



where P(z) and Q(z) are holomorphic for |z f|<|a |. In 
the following section we shall prove that these conditions are 
sufficient. 

COROLLARY. If the integrals at infinity are regular, p(z) 
and q(z) must have zeros at infinity of the first and second 
orders respectively. The proof is left as an exercise to the 
reader. 



87, 88] CONDITIONS FOR REGULAR INTEGRALS 225 

88. The Method of Frobenius. If P(z) and QO) are holo- 

morphic for \z f [< a f|, a fundamental system of integrals 
can be found for the equation 

?(z)div QQ) (1) 



such that both integrals are regular in the neighbourhood of 
If the origin is transferred to f, equation (1) becomes 

Z 2 w'' = 00(2X + \/r(z)w, (2) 

where 0(z) and ^(2) are holomorphic in the neighbourhood of 
the origin. 

Let w = z^c n z n ; then, if $(z)=^faP and \^(z) 



z$ (z) iv \fs(z)w 



n = 

where d = 
and d n = 



Hence, if all the quantities d , d^ d 2 , d B , ..., vanish, and 

o> 
if ^c n z n is convergent, w is a solution of (2). 

u 

The Indicial Equation. The equation in p, 



is called the Indicial Equation. From it can be obtained, in 
general, two values of p. If one of these values is substituted 
for p in the equations d l = 0, c 2 = 0, d 3 = 0, . . . , values for 
c i> C 2> c s> i are found in the form 



,v 
V 



where H n (/o) is a polynomial in /o. 

If the roots of the indicial equation do not differ by an integer, 
none of the coefficients c lt c 2 , c 3 , ... , is infinite. If the roots are 



M.V. 



226 



FUNCTIONS OF A COMPLEX VARIABLE [CH. xm 



p l and p l + m, where m is a positive integer, then when p = p l , 
c m> c m+i> c m+2 are usually infinite. To avoid this we put 
C o = c (p-Pi) which makes c , c lt ..., c m _ 1 , all zero, and c m , c ro+1 , 
c m+2 , ... , finite, when p = p r 

Now assume cZ x = d 2 = c 3 = . . . = ; then 

z 2 w" - Z(f> (z)w' -\!,(z)w = z p c Q {p(p -l)-a p- 6 } , (4) 



Where C n = 

for 7i = l, 2,3,.... 

Let <p(z) and -fy(z) be holomorphic within and on the circle 
|z| = R. Then, if Mj and M 2 are the maximum values of <j>(z) 
and \/s(z) on this circle, 



Thus 
so that, if 

yn = 



+M 



| ^ j 



R" 



then 
Now 



Hence 



Vn 



_ 
\c n 



+M 



R 



Accordingly, if p is finite, and has not any of the values p l 1 , 
p 1 -2, ..., p 2 -l, p-2~ 2 > i wner e PI, p 2 > are the roots of the 
indicial equation, 



-> y 

CO CO 

Thus y]y n z n , and consequently 2 V 1 , converges if 
o o 



THE INDICIAL EQUATION 227 

But if oc is the nearest singularity to the origin, R can always 
be chosen so as to include any point 0, such that | z |<| oc |, within 



the circle. Thus the series ^jc n z n is convergent for | 

co 

and ^v = z p ^ l CnZ n satisfies equation (4), if c lt c 2 , c 3 , ... , are given 



o 
by equations (5). 

Uniform Convergence of the Series with regard to p. Con- 
sider a region K in the p-plane bounded by the large circle 
\p\ = <r and small circles whose centres are those of the points 
p l I, p l 2, . . . , p. 2 1, p. 2 2, . . . , which are interior to this large 
circle. Then, if n = v>cr, for*all points of K, 



Now let v be taken so great that the last expression is always 
positive. Also let M denote the maximum value of 



M 



for the region K. Then, if 



-J R 



we have y n = C n , (n = v, j/ + l, + 2, ...). 

As in the case of the y's, we can obtain 



R' 

CO 

so that 2G H R' U is convergent if R'<R. Thus the series 

71 V 

is uniformly convergent if | z \ = R' and if p lies in K. It is 
therefore holomorphic with regard to both z and p, provided 
that |zj<|tt.|, and that p has any finite values except p l 1, 
p 1 2, ..., /o 2 1, /o 2 ~~2, If, however, p z = p^-\-m and if 
CQ = C(P Pi), the point p 2 m is not excluded. 

The Fundamental System associated with the Roots of the 
Indicial Equation. There are three cases to consider. 

I. Let p l and p., differ by a quantity which is not an integer. 



228 FUNCTIONS OF A COMPLEX VARIABLE [OH. XIH 

Then, if p is equated in turn to p l and p 2 , equation (4) becomes 
equation (2), and we obtain two independent solutions, 



II. Let the indicial equation have two equal roots p = p l ; then 
equation (4) becomes 



If this equation is differentiated with regard to p, it becomes 



-/>i) log s}. 

If in these two equations p l is substituted for p, it follows 
that w and both satisfy equation (2). Thus a fundamental 
system w l9 w 2 , is obtained for equation (2), where 



III. Let /o 2 = /i + m > where ?n is a positive integer. Then, if c 
is replaced by c(p p l ), equation (4) becomes 



Thus equation (2) is satisfied by the fundamental system 



Solutions free from Logarithms. If H m (/o) contains /Q ^ as 
a factor, c can be left unaltered, and both solutions will be free 
from logarithms. In that case w l will be of the form z pl P(z), 
where P(z) is a polynomial of degree =(m 1). 

89. The Gaussian Differential Equation. The equation 



is known as Gauss s Equation, or the Hypergeometric Equation : 
it has singularities at 0, 1, oo . 



88, 89] ASSOCIATED FUNDAMENTAL SYSTEM 229 

CO 

In the vicinity of z = let w = ^c n g f+n ; then 



Thus the indicial equation is 

Xp-l)+yp 

Also, for 7i = 0, 1, 2, 3, ..., 



so that 



There are four cases to consider. 

I. Let 1 y be not an integer. Then, assigning to p the 
values and 1 y in turn, we obtain the fundamental system, 

w l = F(oi > 0, y, z\ w. 2 = z l -yF(oi- 7 +l, /3-y+l, 2-y, z). 

II. Let 1 y = 0. Then the indicial equation has two equal 
roots p = 0. Hence one solution is w l = c Q F(oi, /3, y, z). 

Again, 



= 

so that 

' -l 2 3 



r p+r+l' 

Thus the second solution is 



2 = 

where 

*-' 



III. Let 1 y = ?7i, where m is a positive integer. One 
solution is ^ 1 = c F(oc, /3, y, z). Again, putting c 



i n ~ l 
X 



1 r +-L.)- 



230 FUNCTIONS OF A COMPLEX VARIABLE [CH. xm 

we have 



= ... 

"^ 



Hence the second solution is 

(a m)(oc m + 1) ... (a 1) 



(a m)(ot ?7i-fl) ... (a m + 9i 1) 



{m+n-l/ i 

S v^r~ 



1 1 \ i-l 1 1 1 

-- - 



r r 

If either oc or /3 is one of the numbers 1, 2, 3, ... , m, the terms 
involving log z disappear, and the second integral becomes 

^-^(a-y + l, y + 1, 2-y, z), 

in which the hypergeometric factor is a polynomial. Since p + m 
is a factor of K m (p), this integral could also be obtained by 

o> 

putting p = 1 y in ^c n z p+n . 

o 
Let neither oc nor /3 have any of the values 1. 2, 3, ... , in; 

then, if w 2 is divided by the coefficient of log z, and a multiple of 
w l subtracted from it, the fundamental system can be taken to be 

w 1 = F(a. ) /3, y, z), ^ 2 = io 1 log^ + F 1 (a, (3, y, z\ 
where 

F/a, /3, y, z) 






89] GAUSSIAN DIFFERENTIAL EQUATION 231 

IV. Let 1 y be a positive integer. This case can be reduced 
to Case III. ; for the substitution w = z l ~ *W gives 



where a' = oc y-fl, ' = /# y+1, y' = 2 y; so that 1 y' = y 1 
is a negative integer. 

Thus, if either oc or ft has any of the values 0, 1, 2, ... , y, 
the two integrals are 

F(oc, ft, y, z\ i-vF(oc-y + l, /3-y + l, 2-y, 2), 
where vanishing factors in the numerators and denominators of 
the coefficients of F(a, ft, y, z) are cancelled ; while if neither 
cc nor /3 has any of these values, the fundamental system can 
be taken to be 

, 2-y, z), 

3-y + l, 2-y, z). 
Solutions Regular near z = l. The substitution z = 1 f gives 



Hence solutions regular near = 1 are obtained by replacing 
oc, /3, y, 2 by oc, $ oc + /3 + l y, 12, respectively in the integrals 
already obtained. 

For example, when y oc ft is not an integer, the solutions are 
F(oc, /3,oc+/3+l-y, 1-0), 

(l_3)y *F(y-/3, y-GC, y-a-/3+l, 1-0). 

Solutions Regular at Infinity. If we put 0=1 /f, w = a W, 
then 



Hence solutions regular at infinity are obtained from the 
solutions regular near = by replacing oc, ft, y, 0, by oc, 1-f oc y, 
1+oc ft, 1/z, and multiplying by 0~ a . When a. ft is not an 
integer, the two solutions are 

iu l = z- a F(oi ) 1+oc-y, 1+OC-/3, 1/0), 
m 2 = 0-0F(& 1 + ^-y, l + /3-a, 1/0). 

The Differential Equation of the Quarter Periods of the Jacobian 
Elliptic Functions. If oc = ft = 1/2, y = 1, Gauss's Equation becomes 

iw = Q. (70, Cor.) 



232 FUNCTIONS OF A COMPLEX VARIABLE [CH. xin 

It is left as an exercise to the reader to prove that solutions 
regular near 0, 1, oo , are : 



, , 



i , I 



o^/ter worked examples on differential equations, see 
Chapter XIV. 90, 9L 

EXAMPLES XIII. 

Find regular integrals in the domain of 2=0 for equations 1-16 : 
1. 2zV+2i0'~(l+ a )w=0. 

Ans. *i = '+ + * + * + -' 






2. z(\-z) 

oo fjn 

4. zw" + w' w = Q. An s. u\ = 2 / , \.> > 



... 

5. (l+2)?//-2w = 0. ^?2S. 7^=2 + 2 , W 2 = ?<-'! log 



6. ^' + 2-lw' + %' = 0. 



7. 2^" - li' = 0. ^I?i5. Wj = f) ./ rr-: , 

o w!(^+l)l 

2 2 /2 1\ z 3 /2 2 1 

8, 



89] EQUATION OF JACOBIAN QUARTER PERIODS 23S 

9. 2 V + 0(1-0) w' - ( 1 + 20) w = 0. 

10. ZW" + W' + mZW = 0. A ns. ll\ = 2 ( - 1 )" / t \2o2n ' 



11. 0V + 4?' + 2zfr'=0. An*, i^ l/z, w z = l 

12. 2 l" ' 



13. 2 (l-)?^+2(l-0X-w=0. Ana. ^ 1 

14, ^l -0 



15. 92 2 w"-152w' + (3fo 4 + 7)w=0. ^7w. w x = 1/s cos 2 , w 2 = z lf * sin 2 



1+<V3 l-<y. 

17. Find regular integrals at infinity for 0W+(a 

z 

18. Find regular integrals at infinity for 0%' 

^l?s. %-j = -jj e 1/z2 , w 2 = 2?^ 



[CH. XIV 



CHAPTER XIV. 

LEGENDEE'S AND BESSEL'S EQUATIONS: EQUATIONS OF 
FUCHSIAN TYPE. 

90. Legendre Functions. If the substitution z = l/g is made 
in Legendre s Equation ( 84), it becomes 



Let w=^c v ? +v ; then 






The indicial equation has roots p l = n, p 2 = n + l; and the 
second equation gives c l 0. Also 



(, = 0,1,2,...). 

In the first place assume that p 2 p l or 2n + 1 is not an integer ; 
then, if p = n, 

" --- 4 1 

" 



while, if 






! H L 2 n + I 3 

2 ' 2 ' ^2' 2 



90] LEGENDRE FUNCTIONS 235 

Thus, if 



2 n + 1 HO+l/2)z 



n + 1 



Again, let 2n + 1 be an integer ; then n must either be an integer 
or half an odd integer. 

If n is an integer or zero, all the coefficients are finite. Hence 
both integrals are free from logarithms. In particular, if n is 
zero or a positive integer, 

9/^!\2 

w 'i = c 7^' P "< 2 )' (54, Cor.) 

V*Ji 

If 71 is half an odd positive integer, w 2 is the integral which does 
not involve log z, so that Q n (z) is an integral. If TI is half an odd 
negative integer, w l is the integral not involving log 2. But, in 
this case, since l/r(ii + 3/2) is zero when n + 3/2 is zero or a nega- 
tive integer, the first n terms of Q n (0) vanish, and therefore 

" _* 1_ 
^ 2'2 n ' 

so that Qn(^) is again an integral. 

Accordingly, Q n (z) is an integral for all values of n. It is 
known asLeyendres Function of the Second Kind. P n (z) is the 
more important of the Legendre functions when |z|<l, and 
Q n (3) when!z!>l. 

Note. Thus far P n (z) has only been defined for positive integral 
or zero values of n, while Q M (z) has been defined for all values of n. 

Relation between Legendre 's Equation and Gauss's Equation. 
If in Legendrc's Equation we put z = 1 2f , we obtain 



which is Gauss's Equation with oi = n+l, /3= n, y = 1. Hence, 
in the vicinity of z = l, the two solutions are 



+ 



236 FUNCTIONS OF A COMPLEX VARIABLE [OH. xiv 

Definition of P n (z) for all Values of n. When n is a positive 
integer, 



Now it has just been shewn that this function satisfies 
Legendre's Equation for all values of n. Accordingly, for all 
values of n we define P n (z) by the equation 



COROLLARY. P n (z) = P _ n _ l (z). 

Example 1. If n is zero or a positive integer, shew that 



where the path of integration is taken so as not to pass through the point z, 
[Expand l/(z () in descending powers of z for \z\ > 1, and evaluate the 
coefficients by partial integration. The theorem holds if \z < 1, since the 
functions on both sides of the equation are holoniorphic.] 

Example 2. Use the series for Q n (z) to prove, for all values of n, the 
formulae : 

(i) (n 



Example 3. Use the expression P n (s)=F( -n, n+l t 1, ~^a~) to prove, 

for all values of ?i, the formulae : 

(i) (n + l)P n+l (z) - (2 + 1)P,,() + P M _ 1 () = 0, 
(ii) n-p n (z)=zP' n (z)-?' n -i(z}. 

Example 4. Shew that, for all values of , 
(i) 



[Use Ex. 2, (i), and Ex. 3, (i).] 

91. Bessel Functions. The equation 

z*w" + zw' + (z* - n 2 )w = 

is known as BesseUs Equation, and its integrals are called 
Cylindrical Harmonics or Bessel Functions. 



90, 91] BESSEL FUNCTIONS 237 

The only singularities of Bessel's Equation are z = and z = oo . 

00 

To solve in the vicinity of = 0, put w = z?^c v z v ; then 



v = 

Hence c (yo 2 ^i 2 ) = ; ^{(p + 1) 2 ?i 2 } =0 ; 

c v {(p + vf n 2 } = c v - z , (i/=2, 3, 4, ...). 

The indicial equation is p 2 n 2 = : its roots are p l = n, p 2 = n. 
If pi p 2 is an integer, ??, must either be an integer or half 
an odd integer. The second equation gives c^ = ; so that 
c 3 = c 5 = c 7 = . . . = 0. Also 

r, f IV C 

<"2v \ L ) / 



where v = 1, 2, 3, .... 

There are four cases to consider. 

I . Let n be neither an integer nor half an odd integer. Then 
there are two independent solutions J n (z) and J_ n (z), where 

Z n / Z 2 Z 4 1 

~2 w n(w)l 2(2ri + 2) + 2^27?H-2)(27H-lt)~ "J 



J n (z) is holomorphic for all finite values of 2;, except possibly 
2 = 0: it is known as Bessel's Function of the First Kind of 
order n. 

If n is a positive integer, J M (z) is an integral. J. n (z), 
however, is not a linearly independent integral. For, since 
l/IL(-n + v) = 0, where i/ = 0, 1, 2, ... , ^-1, 



II. Let 7i be half an odd integer ; then, since the coefficients in 
J n (z) and J- H (z) are all finite, these two functions are linearly 
independent integrals in this case also. 

III. Let n = Q, so that the roots of the indicial equation are 
equal ; then 

(-lys" 



238 FUNCTIONS OF A COMPLEX VARIABLE [CH. xiv 

Hence 

co f i \ v 2v v ~[ 

' 



Thus the two integrals are 

J oOO = . 

and v 



Y (2) is called BesseUs Function of the Second Kind of order 



zero. 



IV. Let n be a positive non-zero integer; then, if c Q = 



Hence 

= i^ 



1v 



to + 2)... 

x f Vi 1 , A 1 . y> . 1 :+v_ _i 

\~ l p~n+2r ~{p + n + 'lr ^p + n + ~lr ^p + Sn 

Accordingly, if p = n, 

z n 






z- 1 



z n 



2 2. 4. ..'2n.'2A. ..(2^-2) 



91] RECURRENCE FORMULAE 239 

If these two integrals are multiplied by 2 n ~ l (n l)!/c, they 
become J n (z), and 



(-1)- /W+frf * 1 






Subtracting ;;( y + ^ + H j) Jn(z) from the latter integral, 

we obtain the integral, 

Y M (2) = J(z) log z - ^ V (ll ~~ v ~ 



where ^r)i|++...+ v (r-l l 2, 3, ...), and 

Y n (z) is called Bessel's Function of the Second Kind of order n. 

Recurrence Formulae. We leave as an exercise to the reader 
the verification of the following formulae : 

(i) 2J n '(s)=j n . 1 (2)-J n+1 (2); 

(ii) J '()=-J 1 (); 



J w () as a Function of n. Let 2 1 = R, | w | = N ; then, if m is 
an integer such that m N> 1, and if 



T ,v 



00 

then | T v (z) \ = M p . But 2 ^ ^ s convergent ; consequently, by 

m+l 

o 

Weierstrass's M Test, 2 ^(2) is uniformly convergent if 






240 FUNCTIONS OF A COMPLEX VARIABLE [OH. xiv 

\ 

Now R and N can be chosen so large that these regions enclose 
any assigned finite points z and n. Accordingly, for all finite 
values of z, except possibly 2 = 0, J n (z) is a holomorphic function 
of n. 

The Bessel Function G. n (z).* It is sometimes found convenient, 
instead of J_ w (z) or Y n (z), to take as the second solution of 
Bessel's Equation the function 



where the limiting value of the expression on the right-hand 
side is taken for G n (z) when n is an integer. 
Now 



Also 

PW+8l TYn vs 



, v _ 
v^4i/!r(- 

so that 



/ 

v ~~ - 1 ; I o 



7T . VI 

-n+2v 

^(-n + v). 



If 7i is a positive integer, let p = n; then 



Accordingly, if n is a positive integer, 

O f T / \ : -r , 



2 COS 717T 



*Cf. J. Dougall, Proc. ^IM. Math. Soc., Vol. XVIII. p. 36. 



91] THE FUNCTION G n () 241 

The verification of the following formulae is left as an exercise 
to the reader : 

(i) G_ tt () = e**G M (z); 

(ii) 2G n '() = G n _ 1 (*)-G n+1 (); 

27? 

(iii) ^G.(z)=G,,. 1 (*)+G l , +1 (X 

THEOREM. If P(z) and Q(z) are any solutions of Bessel's Equation, they 
satisfy a relation of the form 



where C is a constant. 

For, if the substitution iv=z~ l ' 2 W is made in Bessel's Equation, it becomes 



Consequently 

{x 

Hence, integrating, we have 



a * - F() Q(*)= j- 
For example, 



1 sin mr . 

T(n) = - 



and therefore J n ( 2 ) j'_ n (z) - J' n (z) J_(,-)= - 2 

The reader can easily deduce that : 



~J(*)J-ti-lW sillWTT. 

.=2 



(iii) G n 



The Zeros of J n (s). If n is real and greater than - 1, all the zeros of J n (z) 
are real and distinct, except possibly z=0 ; this can be shewn as follows. 

We have - 



Thus, multiplying the first equation by J n (fiz), and the second by J,,(ou), 
and subtracting, we have 



242 FUNCTIONS OF A COMPLEX VARIABLE [OH. xiv 

Hence, if K(?0> -1, 

(<X 2 - 

Therefore, if O=OL and 6 ft are distinct zeros of J M (0c), 



Again, let /3=cx + ; then 



If this equation is divided by e, and is then made to tend to zero, the 
equation becomes 



Hence, if 0=oc is any zero of J M (#c), except ^=0, 



THEOREM I. If n is real and greater than -1, J n (z) cannot have any 
purely imaginary zeros. 



For 



and the latter expression cannot vanish if y is real. 

THEOREM II. If n is real and greater than -1, J n (z) cannot have a 
complex zero. 

For if z=p + ig is a zero, where p and q are real, z=p iq must also 

be a zero ; hence 

P- 

I x J n { (p + iq)x } J n { (p iq)x}dx=Q. 
Jo 

But if n and x are real, the integrand is positive ; and therefore the 
integral cannot be zero. Thus the theorem must hold. 

Accordingly, if n is real and greater than 1, it follows that every 
zero of J n (z) must be real. 

THEOREM III. If n is real and greater than - 1, 3 n (z) has no repeated 
zeros except possibly 2=0. 
For if 2=0. is a zero, 



so that J'(o-) + 0- Thus Jn() has no repeated zeros. 



91,92] THE ZEROS OF J H (z) 243 

THEOREM IV. If n is real and greater than 1, J n () and J n +i(z) have 
no common zeros except possibly 2 = 0. 
This follows from the formula 



92. Equations of Fuchsian Type. Equations whose coefficients 
are meromorphic in the entire plane, and which have their 
integrals regular in the vicinity of all their singularities, are 
called Equations of Fuchsian Type. 

If the singularities are a lt a z , a 3 , ..., a n , and infinity, the 
equation is of the form 

dw P 2n . 2 (z) 



= 
dz* (z - Oj)(z - 2 ) ... (z - c^) dz^(z- a l ) 2 (z - a 2 ) 2 . . . (z - a n 



where P n _ 1 () and P 2 n-2( 2: ) are polynomials of degrees n 1 and 
Zn 2 respectively ( 87). 

If infinity is not a singularity, the equation is of the form 



dz* (z-a l )(z-a 2 )...(z-a n ) dz(z-a 1 )\z-a 2 )\..(z-a n ) z 

where the coefficient of the highest term in P n -i(z) is 2 
(83, p. 212). 

THEOREM. The sum of the indices associated with the 
singularities a lt a 2 , ... , a n , oo , of the equation of Fuchsian Type 
is Ti- 1. 

Let P_z 



and let ^ (z) (z a t ) (z a z ) . . . (z a n ). 

Then the indicial equation for the singularity a r is 

p(p 1) = p ""^ ^-\- terms independent of p. 
\js \a r ) 

Accordingly, if the roots of the indicial equation are p l and 



Now, by the theory of partial fractions, 



244 FUNCTIONS OF A COMPLEX VARIABLE [CH xiv 

Hence, integrating round a large circle which encloses a lt 
a 2 , ... , a n , we have 




since Lim 2 

s^so 

Thus the sum of the indices at a lt a z , ... , a,,,, is n + A. 
Again, put z=l/f ; then the equation becomes 



Thus the indicial equation is 



so that p l + /o 2 = 1 A. 

Hence the sum of the indices is n 1. 

COROLLARY. If infinity is not a singularity, A =2, and 
therefore the sum of the indices is n 2. 

93. Riemann's P-function. We shall now investigate the 
conditions that the equation 



should be completely determined if the n + 1 singularities 
a lt a 2 , ... , a n , oo , and the corresponding indices, are assigned. 

There are 3n l constants to be determined in the equation. 
The assigning of the singularities a l} a 2 , ... a w , oo , simply deter- 
mines \fr(z) and the degrees of P w _!(2) and P 2n _ 2 (z). The assign^ 
ing of the 2rz, + 2 indices determines only 2n+I constants, since 
the indices must satisfy the condition that their sum is nl. 

Thus n 2 constants remain to be determined ; so that, if 
7i = 2, the equation is completely determined. 

Similarly, when infinity is not a singularity of the equation, 
there are n 3 constants to be determined ; so that the equation 
is completely determined if n = 3. 

Consequently, in both cases, if there are three singularities, and 
if the indices are given, the equation is completely determined. 

By means of the transformation 

zhcb a 



92, 93] RIEMANN'S P-FUNCTION 245 

the equation with singularities h, k, oo , can be transformed into 
an equation with singularities a, b, c. The equation can therefore 
always be put in the form 

d 2 w ( f g h }dw 



where f+g + h = 2. 

Let the indices at a, b, c, be X and X', /UL and //, v and j/, 
respectively, where X + X' + yu + yu' 4-^ + ^=1. Then, since the 
indicial equation at a is 



1 +/= x + X', l=- \\'(a - b)(a - c) ; 
so that /=X + X'-1. 

Similarly g = /m + // 1 , m = /*//(& ~ c )(^ ~ a ) 
h = v+i> 1, w= i/i/(c a)(c 6). 
Hence the equation can be written 






/uL /u.'l vv'\ dw 
~~ ~ 



dz*^\ z- z-b z-c J dz 

+ \~ z-a ~ +J z-b ~ + z^c~ 

w 



(z-a)(z-b)(z-c) 
Now for simplicity assume that X X', /U. JUL', v v, are not 
integers ; then, if P A , Py, P M , P M ', P,,, P v >, are integrals corresponding 
to the indices X, X', /UL, /UL', v, v', any branch of any integral of the 
equation can be expressed in any of the forms 



Riemann denotes such a function by 

{a, b, c, \ 
X, yu, v, z V; 
X', yu', I/', J 

and it is called Riemann s P-function. If either X and X', 
yu and //, or i/ and i/', are interchanged, the differential equation 
remains unaltered. Likewise the three columns can be inter- 
changed without altering the equation. Again, if the function 
is multiplied by (z a)*(z c)*(z 6)-" p , the indices at a, b, and 



246 FUNCTIONS OF A COMPLEX VARIABLE [CH. xiv 

c, become X + or and X' + a; /JL <r p and // o- /o, v + p and i/ + p, 
while the branches of the function remain holomorphic at all 
other points, including infinity. Also the sum of the indices is 
still unity. Consequently 

a, 6, c, } I a, b, c, 



\ x, M , i/, z V=P^ x+cr, M-O p, j/+/o, z y. 
\ / / / i ^ / - / / , 

X , p , v , J ^X+o-, /m <r p, v +p, 

Again, the transformation 

f_za c b 
^~ z b c a 

changes a, b, c, into 0, oc , 1. When the latter three points are 
the singularities of the equation, the function is denoted by 



p/X, M, , z \ 

IX, JUL , V, 



where K is a constant ; thus 

Z '(i-*y?{ x ' * ": .j-pf^*' "C'l* "-?' 4 

tX, /x, i/, J LA+O-, /x o- p, ^+/>, 

The differential equation determined by P( J M ; ^ 2 1 is ob- 

IX, M, j/, J 

tained by putting a = 0, c = l, and making b tend to infinity; 
it can, by means of the equation X-f-X'-f A* +/*'+" -fV^lj be 
put in the form 



XX' - (XV ^ 



In particular, the function P( 1 ' ^' 3 2) satisfies 

! . \1 y, P, y . p, / 

the hypergeometric equation 



Note. Since 

' K 

-y, ft y a ft 

where a = X + /* + ^ /3 = X + /x r + 1/, y = l-X x + X, it follows that 
the P-function can always be expressed in terms of the integrals 
of the hypergeometric equation. 



93] INTEGRALS OF HYPERGEOMETRIC EQUATION 247 

The Twenty-four Integrals of the Hypergeometric Equation. 
The solutions corresponding to the indices 0, 1 y, at z = are 
(89), 

F(OC, ft, y, 2), Z l -y~F(oL y + 1, /3-y+l, 2-y, 2): 

we denote them by W 1 (0) and W 2 (0) respectively. 

Alternative forms for W 1 (0) and W 2 (0) are obtained as follows. 

We have 

/ 0, oc, 0, 

0, y 8, GC + /3 y, \ 
_ y> y _ a> 0, '/ 
Thus 

(l_z)Y--0F(y-a, y-/3, y, 2) = C t W^ + C 2 W 2 (0) . 

But, since the function on the left-hand side is uniform at 
2 = 0, C 2 = ; hence 

In this equation let 2 = ; then 1 =C r Therefore 
W^ = (l-z)y-*-PF(y-OL, y-/3, y, 2). 
It follows that 

In like manner alternative forms can be found for the regular 
integrals at infinity and 2=1. 
Again, the six transformations, 

when applied any number of times in any order, change the 
points 0, oo , 1 , into the same three points in different orders. 
By means of these transformations new forms can therefore be 
obtained for the integrals. For example, 

jg 
0, 0, oc, 



0> a> 0> 

v , y-ft/S- 
= C(l-z)-F(a, y-/3, y, ^J. 



248 FUNCTIONS OF A COMPLEX VARIABLE [CH. xiv 

In this equation let z = ; then C = 1. Hence 



These two expressions for W/ ) are valid if R(z)<l/2. 

We have thus obtained four different forms for W/ ). Similarly 
four different forms can be found for W 2 <>, W^ 1 ), W 2 < ] >, W^ 00 ), 
W 2 <*>. These twenty-four forms for the integrals of the' hyper- 
geometric equation are : 



III. 

IV. 

V. 
VI. 

VII. 

VIII. 

IX. 

X. 

XL 

XII. 

XIII. 
XIV. 

XV. 

XVI. 

XVII 



2-y, z) 



a-y + 1, 



z-l 

z 
z-l 

z 



a-y + 1, a- 



93, 94] ANALYTICAL CONTINUATION 249 

xvm. =^-*(- 

XIX. = z-l' 



1-z 
XX. = z l ~ 

XXI. W 2 (> = 2-fl 

XXII. =2 a - Y (z-l) Y - a - p F( 1 -a, y-a, - 

XXIII. = ( l)-*Fnft y-a, /3-a+l, y-^ 

Y Y T^r 1 ~ V / l\V~^~lTTI/3 _1_ 11 rv /Q rv _L 1 

\ * 21 

Relations of the form 



where r=l, 2; s = 0, 1, oo; ^ = 0, 1, oo ; 

hold between the six functions W/ ), W 2 ( ), W^), WgW W^"), 
W 2 ^\ One of these relations is given in Example 4 of 63, and 
the others can be found by similar methods. 

Example. Shew that, if y -.-/? is not an integer, the analytical con- 
tinuation of F(OL, /?, y, z) in the vicinity of z = \ is 



[Apply Ex. 4, 63, to form III. of W^.] 

94. Spherical Harmonics. The equation 



is called Legendre's Associated Equation. The integrals of this 
equation are called Spherical Harmonics of degree n and rani- m. 
The most important cases are when n and in are positive integers, 
such that in =11. 

If 771 = 0, the Harmonic is a Legendre Function or Zonal 
Harmonic. 

If ?7i = l, 2, 3, ... , 7i 1, the Harmonic is a Tesseral Harmonic. 

If ?7i = 7i, the Harmonic is a Sectorial Harmonic. 



250 FUNCTIONS OF A COMPLEX VARIABLE [CH. xiv 

Let the substitution w = (z 2 1)* MI W be made in Legendre's 
Associated Equation ; then 



Again, differentiating Legendre's Equation m times, where m 
is a positive integer, we obtain 



Accordingly, if m is a positive integer, two independent solu- 
tions of Legendre's Associated Equation are 

-r/ m P (?\ m rlmC\ (y\ 

p m(~\ _ / 2 2 _ 1 \ 2 a r n\Z) r\ w / \ _ / 2 i \2 a ^>n\Z) 

dz m ' ^ n ( ' dz m ' 

These functions P n m (z) and Q n m (2) are known as Legendre's 
Associated Functions of the First and Second Kinds respectively. 
To make them uniform a cross-cut is taken along the real axis 

in 

from oo to +1, and that branch of (z 2 1) is chosen which is 
real and positive when z is real and greater than 1. 
If m and n are positive integers, and m = n, 

- 120 > 



( 93, Form II. of W 

(7^ + m)! /2-i\i / 1_ 2 \ 

= ,/ \I'("TT) F Ti+1, w, ?n + l, ^r 
77i! 71 771! \2; 4-1 / \ 2 / 



If m>7i, P n m (z) = 0. 

Similarly, if in is a positive integer, then for all values of n, 
_(-iy*( z 2 
+" 



M + m+2 ii + m + l 31 



941 LEGENDKE'S ASSOCIATED FUNCTIONS 251 

Again, let the equation 



obtained from Legendre's Associated Equation by means of the 
substitution w = (z* l)- * W 

be differentiated m times ; then 



Hence, if 971 and n are positive integers, and if m = ti, two 
independent solutions of Legendre's Associated Equation are 



and 



771 f 3 f 2 f z 

= ( 2 ; 2 -l)-2 ... 

J OO J <X> J 00 



Since the four functions P n m (z), P~ m (z\ Q n M (z), Q; m (2;), satisfy 
the same equation, they cannot all be independent. The relations 
connecting them are found as follows : 

"1 m flnm 

f=-W- s r a ('-) 'Sa^-l)- 



" l ; * 



3 1 



(93, Form II. of W 



252 FUNCTIONS OF A COMPLEX VARIABLE [CH. 

EXAMPLES XIV. 

1. Shew that, for all values of n : 

(i) 
(ii) 

(iii) 

(iv) 

2. Shew that 

(i) Qo(*) 
[Use Ex. 1, 90.] 

3. If n is zero or a positive integer, shew that positive circuits about 
z=l and z= -1 decrease and increase Q,(2) respectively by Tn'P^). [Use 
Ex. 1, 90.] 

4. Use the formula of Example 1, 90, to prove the formulae of Example 2, 
90, for positive integral values of n. 

5. For all values of n, shew that 

(* ~ l){Qn(*) Pn(t) ~ P n (z) <&(*) } = C, 

where C is a constant. [Substitute P n (z) and Q,,(z) for w in Legendre's 
equation, multiply the two equations so obtained by Q n (z) and P n (z) respec- 
tively, subtract, and integrate.] 

6. If n is a positive integer, shew that 



7. If n is a positive integer, prove : 

(i) n{P n (z)Q n ^(z)-Q n (z)? 
(ii) 

8. Shew that 

(i) z3n 
(ii) zG n 

9. Prove that 

(i) 
(ii) 



10. Shew that: (i) J^z)^ sinz ; (ii) J_^(z) 

Deduce that, when n is half an odd integer, J n (z) can be expressed in 
terms of elementary functions. 

11. Shew that : 

(i) ~{z n 3 n (z)} = z n 3n-i(z) ; (ii) {z^ n (z)}^ -z~ n J n+l (z) 



xiv] EXAMPLES XIV 253 

12. Shew that 

,]m 
(i) 2' ?L J H (Z) = C J-(*) ~ C, J- m + 2 + -.+(- 1) M C W J, l + m (4 

(ii) 2- ^ G ()= CoG^s) - cA^+oCO + ...+(- l)"^G, i+m (4 
where c , c 1} ... , c m , are the coefficients in the expansion o 

13. Establish the expansions : 

(i) ?J n - 1 (z) = n,J n (z)- 



14. Shew that J n (z) is the coefficient of f" in the expansion of e ^ f* 
in powers of ^. 

15. Establish the expansions : 

(i) co8(am^)=J 
(ii) sm(2sin^) = 2 
[In Ex. 14 put f=e ifl in turn.] 

16. If n is a positive integer, prove 

J n (z) = - rco$( 

IT JO 

[Multiply expansions (i) and (ii) of Ex. 15 by cosw^ and sinnO, and add.] 

17. Shew that ' 



- __ 

[Multiply together the expansions of e \ *' and e V f /, and find the 
term independent of .] 

18. If R(w)> - J, shew that 



[Expand cos(2cos<^>) in powers of 0, and evaluate the coefficients.] 

19. Solve 2?0"+w=0. 

Ans. 

20. Solve 2%"-2 



21. Solve 



22. If i, w, A % , are positive integers, and k<m, k<n, shew that 
(i) wl 



254 FUNCTIONS OF A COMPLEX VARIABLE [OH. xiv 

23. If n is an integer, shew that 



Equate the coefficients of f in 
24. If n is an integer, shew that 



25. Deduce Gauss's Theorem ( 61) from the Example of 93. 

26. Shew that, if y - a. - /?<0, 

F(a, ft y, *)_r(. 

Sf <i-,r-'~ r 

while if y a. )8=0, 



[For the second equation apply Ex. 4, 63, to Form III. of W/' ( 93).] 

27. Shew that, in the domain of the origin, every solution of Legendre : s 
Associated Equation can be put in the form 



CH. xv. 95] 



CHAPTER XV. 

SOLUTION OF DIFFERENTIAL EQUATIONS BY DEFINITE 

INTEGRALS. 

95. First Method of Solution.* If Q(z) and L(z) are quadratic 
and linear functions of z respectively, and K is a constant, the 
equation Q 

can be put in the form 






w = 0, (A) 

where R(z) is linear in z. We shall confine ourselves to the case 
in which the factors of Q(z) are distinct. 

If the function I 0(f)(f z) A+1 fZf is substituted for w in 

Jc 
equation (A), then 



f 



te=; 



so that I #(^){X(f ) x " 1 Q(f)+(f 2^E(f)}df=sO, (B) 

Jc 

Accordingly, if 0(f) satisfies the equation 



equation (B) becomes 



Jordan, Cours d' Analyse, t. in, p. 240. 



256 FUNCTIONS OF A COMPLEX VARIABLE [CH. xv 

Xow, equation (c) gives 







where p and q are constants, and f a, f &, are factors of 

Thus 

so that 

where D is a constant. 
Accordingly, 



is an integral, provided that either (f )*( &) 9 (f zY vanishes 
at both extremities of C, or else C is a closed curve such that 
this function (or the integrand) has equal values at the initial 
and final points. 

Let P be any point of the f -plane, and let A, B, and Z, 
denote loops drawn positively from P about a, 6, and z. Also 
let A, B, Z, denote the values of the integral 



taken round these loops, with M as the initial value, in each 
case, of the integrand at P. Any of the contours ABA" 1 B~ 1 , 
AZA- 1 Z~ 1 , BZB~ 1 Z~ 1 , where, for instance, the first denotes the 
loops A, B, A" 1 , B - 1 , described in succession, can be taken as path 
of integration C. For, if ABA~ 1 B~ 1 be taken, the final value of 
the integrand is equal to its initial value multiplied by 

^irip glniq ^ - Snip g - 2niq == J . 

and similarly with the others. 

Let the values of the integral taken round these three contours 
be denoted by [AB], [AZ], [BZ], respectively. The value of 
[AB] can be found as follows. 

The loop A gives the integral A, and brings the integrand 
back to P with the value Me 2>rip . Thus the loop B gives the 
integral e* wip B, and the final value of the integrand is e^^+^M.. 
After describing the loop A~ l , the final value of the integrand 
is e 27 " 5 M, so that the corresponding integral is e 2niq A ; similarly 
the integral due to the loop B' 1 is B. Thus 
[AB] = (1 - e 2 ^) A - (1 - 



95] 



BRANCH POINTS OF THE INTEGRAL 



257 



Similarly [AZ] = (1 - e 2 -*) A - (1 - 
and [BZ] = (1 - e 2 ^)B - (1 - 

Hence (1 -e 2 ^)[AB] + (l -e 2 ^)[BZ] + (l - e 2 ^)[ZA] = ; 

so that a linear relation exists between the three integrals, as is 
to be expected. Any two of these integrals, say [AZ] and [BZ], 
can be taken as the fundamental system. 

The Branch Points of the Integral. When z is fixed, the path 
of integration can be deformed without altering the value of the 
integral, provided that it is not made to pass over any of the 
points a, 6, z. If z varies continuously, the integrals will also 
vary continuously, provided that the path of integration is 




Fio. 76. 



deformed, when necessary, so as to avoid passing through the 
points a, 6, z. 

If z describes a contour about a, the loops A and Z (Fig. 76) 
must be deformed into loops A' and Z'.* 

Now Z' is equivalent to ZAZA^Z' 1 and A' to ZAZ" 1 or 
ZAZ~ 1 A~ 1 A. Thus, if Z' and A' are the values of the integrals 
taken along Z' and A', 

A'=-[AZ] + A. 



* This can be effected as follows : (i) deform Z into Z lf so that z passes round a 
to z l ; (ii) deform A into A' ; (iii) deform Zj into Z', so that z moves from % into its 
original position. 

M . 1 '. R 



258 FUNCTIONS OF A COMPLEX VARIABLE [CH. xv 

Accordingly, [AZ] is transformed into [AZ]', where 
[AZ]' = (1 - e***)A? - (1 - e^p)Z' 



(D) 

Similarly [BZ] becomes [BZ]', where 

[BZ]' = [BZ] + (e 2 ^ - l)e 2 *[AZ]. 

Thus a is a branch point of both integrals. Similarly it can 
be shewn that b is a branch point. Infinity is also, in general, a 
branch point ; but a circuit about it can always be replaced by 
circuits about a and b. 

96. Gauss's Equation. If in equation (A), 95, 

Q(z) = 2-z 2 , R(3) = (a-y + l)-(oc.-0 + l)2, X=-a-l, 

then a = 0, 6 = 1, > = . y + 1, # = y /3; 

thus the equation becomes Gauss's Equation, 



and has the integral 



where C is so chosen that the initial and final values of the 
integrand are identical. 

A second integral can be obtained by interchanging a and /3, 
and a third by putting 1/f for The latter integral is 



Employing the notation of 62, we can write one such integral, 



f 



where the initial point lies on the real axis between and 1, and 
the initial values of ^~ 1 and (l-f)v-^- 1 are real and positive. 
If z describes a closed contour enclosing z = but not z = 1 , the 
singular point 1/2 will describe a closed contour enclosing z = 
and z = l; and therefore the contour of the integral need not 
altered. Accordingly, for values of z which lie in a simply- 
connected region enclosing 2 = 0, but not enclosing 2=1, the 
integral is a uniform function of 2. 



96, 97] THE HYPERGEOMETRIC FUNCTION 259 

Now let |z|< 1, and choose that value of (1 z )~* which has 
the value + 1 when z = ; then 

,0+, 1-,0-J 



^ 

)} 
-)2 w , (62) 



, y-/3)F(oc, y, 2). 

Note. The expression given by this equation for the function 
F(oc, /3, y, 2) as a contour integral is valid for all values of z. 



Example. Prove F(o., /?, y, 2) = (1 - z) ~ a F (OL, y - /?, y, 

" [Put = l-C.] 
Again, consider the integral 

where the initial point is on the straight line joining f = to 
f =z, and the amplitudes of f/2 and (l f/2) are taken to be zero 
at this point; while that branch of (1 f)?"^ 1 is taken which 
has the value 1 when f =0. From formula (D) of 95 it 
follows that when 2 describes a closed contour about 2 = 0, the 
integral is multiplied by e~ Zvi y. 

Now let =zZ; then the integral becomes 

fO+,.1 + ,0-,1-) 
Z a ~v(l Z)~ a (l zZ)?-' 



, a-y + 1, 2-y, 2). 
This equation gives an expression for the function 

zi-yF(oc-y+l, /3-y + l, 2~y, 2), 
which is valid for all values of 2. 

97. Legendre's Associated Equation.* If in equation (A), 95, 

Q(z)=l-z 2 , R(z)=- 2(w + 1)2, \=- w -m-2, 
then a = 1, 6 = 1, p = ^ = w- + 1 ; 

*Cf. Hobson, Phil. Trans., Vol. 187. 



260 FUNCTIONS OF A COMPLEX VARIABLE [OH. xv 

thus the equation becomes 

(l-z 2 )i</'-2( 
and has the integral 



f (f*- 

JC 



where C is a suitable contour of integration. Hence ( 94) 



is an integral of Legendre's Associated Equation. 
The Function P n m (z). Consider the function 



f(z+, 

=(z 2 -i)H 



where a cross-cut is taken along the real axis in the z-plaiie from 
1 to x to make the function uniform in z, and the amplitudes 
of zl and z+1 lie between TT and +TT. Let A (Fig. 77), a 
point in the f -plane on the straight line joining f =1 to f=z, be 
taken as initial point; and let the initial amplitudes of f 1 and 
f+1 be and 0', where these are the angles (between TT) 




which the lines joining f=l and f = 1 to A make with the 
positive -axis. Also let the initial value of amp(f z) be 
-.(TP ,^ so that amp(f z) is zero for points on the contour 
at which f z is a positive real quantity. Thus if z lies on the 
cc-axis to the right of +1, the initial values of amp(-l-l), 
amp( 1), and amp (f z) are 0, 0, and TT, respectively. 



97] THE FUNCTION P n m (z) 261 

Now let f 1 =(z 1)Z ; then the initial value of amp Z is zero. 
Again f+1 = 2U + ^-ZJ. But when f=l, amp(f+l) = 0; 

hence amp (l -4-?-^ Zj is zero when Z = 0. Also 



where amp(l Z) is initially zero. Thus 

('?-!_ 1 \i-l 
IJL ) 2 
3 I/ 






fl+, 0+,1-,0-) / 'yl \w 

Z(1-Z) ^l + ^Zj 



/^__1\T( 1 +. o+, i-,o-) 

\ y 






In particular, if 771 = 0, 

rjH-, !+,-,!-) 
(f 2 l)(f 3)-*- 1 df=2 w+2 7re' wl sin?i'7r 

1-^ N 



so that ( 90) 



Now, if ?7i is a positive integer, then ( 94) 



2 __ iiH 



r( Z +,i+,2-,i-) 

J. (f 2 - 



262 .FUNCTIONS OF A COMPLEX VARIABLE [OH. xv 

But this function satisfies the differential equation for all 
values of ra. Hence, for all values of n and m, P n m (z) can be 
defined by any one of the equations 



i _ 

(93) 



COROLLARY. P? n _ ,(z) = P(z). 

Example 1. Shew that 

(Z+, 1+, Z-, 1-) f(Z+, 1 + ) 



/ 



f(z+, 1+) 

deduce that 



Function Q, n m (z)> Again, consider the function 
~ l 



where a cross-cut is taken along the real .axis in the z-plane 
from 1 to oo to make the function uniform in z. Let the 
origin in the f -plane be taken as initial point ; and let +1 and 
1 have initial amplitudes 2?r and TT respectively, so that 
they will both have amplitude zero when f is real and greater 
than 1. Also let the initial value of amp (f 2) be ampz TT. 
Then, if | z 



l Sin 717T 



/ ,2r+3\ 
2 / 



(Exs. VIII. 7) 



) 

t , , ( 62,Ex. 2). 



97] THE FUNCTION Q H W (z) 263 

Now, if m is a positive integer ( 94), 
O ^-(- 



But we have just shewn that this function satisfies the 
equation for all values of n and in. Hence, for all values of 
n and m,'Q w m (z) can be defined by either of the equations 



_ 

< - 



vli ,/w+m+2 

XN -I- 1 < 



2 n+] 



f(-l+, 

X ) 



COROLLARY. By applying the formula ( 93) 

F(a, ft, y, ) = (!-)' F( y -a, y-ft y, f). 
we obtain the relation 



A Second Expansion JOT Q n m (2). Consider the function 



There are two cases to consider, according as I (z) is positive 
or negative. 

Let A (Fig. 78), the initial point, be on the straight line 
joining f=l to =z, and let this line make an angle 
with the positive -axis. Also let the initial values of 
amp(f+l) and amp(f z) be and (TT 0) respectively. 
Then if f+l=(z+l)Z, the initial value of ampZ is zero. Also 
f z = (z+l)(Z 1), so that the initial value of amp(Z l)is TT. 

Again, since f 1= 2(1 -- ~^\ an d since, when f = 1, 

\ 2t J 

1) has the value TT in the first case, and the value TT 



264 



FUNCTIONS OF A COMPLEX VARIABLE [OH. xv 



in the second case, -1 has the value 2e iir l--- -Z) when I(z) 

Lt 



is positive, and the value 2e~ iir (l -- 9^ z ) when I(z) is negative. 



-i 




FIG. 



Hence the given function has the value 

e niri ~~ m 



\z+l/ 

fi+,o+,l-,o-) 
Z n (I-Z)- n - m - 



dZ 



m) 

x F^-TI, Ti + 1, 1 -m, ^t 2 ), ( 96), 

according, as I(z) is positive or negative. 

Now let L, M, N, be the values of f(f 2 - : l) n (f-2)" w " m ; 1 d^ 

taken round loops from f=0 about 1, 1, z, respectively; the 
initial value of amp( 1) will be TT or TT according as I(z) is 
positive or negative. Then ( 95) : 



C(z+ , 1+, 

p.,, 

f(-l+,+ 



97] 



A SECOND EXPANSION FOR Q n ' n (z) 



265 



Denote the integral in the last equation by W l ; the initial 
value of amp(f 2 1)" in this integral is mri, according as I (2) 
is positive or negative. Again, let W 2 denote the integral 

1 > ({ 1 -i) r tf-*)**' r *' l 4t in which the initial values of 
fc.rnp(f+l) and amp(f 1) are 2?r and TT respectively; then 
an:p (f 2 1 ) n is mr initially. Hence 

e llni W 2 = e^ niri W l ; 

so that W 2 = e-' l7r ^ n7ri (L-M), 

according as I(z) is positive or negative. 



But 



Hence, since 



1 ~ 



fz+, 



2*+! 






it follows that 

O m/ 2 \ = _ 



x 



e ^m 



according as 1(2.) is positive or negative. 
COROLLARY. From the equation 



it follows that 



(c . p . 263) 



-(J.tl)'- F (-,,.,, +I , 1+ ,,!J-"). 



Example 2. Shew that 



266 FUNCTIONS OF A COMPLEX VARIABLE [OH. xv 

98. Second Method of Solution. Differential equations of 
the type 

(az+a')w"+(bz + b')w' + (cz + c')w = Q (A) 

can be integrated as follows : 

Substitute w= 0(0 6 ^f i n equation (A) : then 
Jci 

( <l>(S)e*{(a^+b+c)z+(a'?+b'S+c')}dS=(). (B) 

JC 

Hence, if <p(g ) satisfies the equation 

(a'? + b'+c')<t,(f) = ^{(a?+bS+c)<t,(t)}, (c) 

/* .^7 

equation (B) becomes jf.O()d=Q, 

Jc <* ~ 

where 0(f)#(f)(a*+ &+ e), Also equation (c) gives 



Thus <J>()e^ z dg is a solution of equation (A), provided C is so 
Jc 

chosen that 6(z) regains its initial value at the final point. 

99. Bessel's Equation. In Bessel's Equation ( 91) put w = z n W; 
then 



This is an equation of the type considered in the previous 
section. Accordingly, since, in this case, 



=f 

J 



W 

c 



is an integral, provided (9(z) or e*(f 2 + l) w +* regains its initial 
value at the final point. 

Hence, if f is replaced by ig, a solution of Bessel's Equation is 



'c 
where C is a suitable contour. 



98, 99] EXPRESSION FOR 3 n (z) 267 

Expression for J n (z). Consider the integral 

[-i+, +i-) 



where the initial point lies on the -axis between 1 and +1. 
Let the initial amplitudes of f+1 and f 1 be 2-Tr and TT 
respectively, so that each of them has zero amplitude at the 
point where f crosses the f-axis to the right of = 1. Then 

f(-l+,+l-) /A z \v fl 

*(?- !)"*#= V J 

J v=0 

= - g* cos ^r(^ 




(Exs. VIII. 7) 
(62, Ex. 2) 



Hence 



COKOLLARY. If 



Example. Prove 



Expression for G n (z). There are two cases to be considered. 
CASE I. Let 7r/2 = = 7r/2 , where <j> = amp z. 
Then consider the integral 



taken along the contour C of Fig. 79 from infinity back to 
infinity. Both extremities of C approach infinity in a direction 
making an angle ?r/2 with the positive -axis ; so that iz is 
real and negative, and therefore 6(g) tends to zero at both 



268 



FUNCTIONS OF A COMPLEX VARIABLE [OH. xv 



extremities. The amplitudes of f+1 and f 1 are chosen so 
that they vanish at the point L, where the curve crosses the 
positive ^-axis. 

If necessary, deform the path so that, at every point on it, 
; then 




-1 O +1/L 



FIG. 79. 



Now put X = e^zg, so that the initial and final values of amp X 
are and 2?r respectively ; then if 






W = : 



where the integral is taken along the contour of Fig. 60 (61). 
Hence 



( 62, Ex. 2) 



Thus J_.(,)- 



99] EXPRESSION FOR J_(z) 269 

Next, let R(> + i)>0, and - Tr/2 < ^ 7r/2 ; then/deforming 
the path C into a contour (Fig. 80) consisting of a line through 




FIG. 80. 



f=l, which makes an angle ?r/2 with the positive -axis, 
described from x to 1, the -axis from 1 to 1, and this path 
reversed, we have 



S^ !>-*" 



where in the latter integral f 1 and f+1 have the amplitudes 
corresponding to the first description of the line from oc to 1. 
Hence 



so that 

G M (z) = e 8niri cos nTr 






Now let f l = e ^ 27r4 ' X, so that X is real and positive; then 
since, when f=l, amp(f+l)= 2?r, 



270 FUNCTIONS OF A COMPLEX VARIABLE [OH. xv 

Thus 



- 



e- 



CASE II. Let 7r/2 ^ ^ 37T/2. 
Consider the integral 



f 

J 



where C is the contour of Fig. 81, and the amplitudes of f 
and f -|- 1 are chosen to be zero at L. If | f | > 1, 



yy iy 

( } 




FIG. 81. 



so that, applying the transformation X = 
we obtain in the same way as before, 



to this integral, 



Next, let RO + J) > 0, and 7r/2 ^ < 3-7T/2 ; then, deforming C 
into a contour (Fig. 82) consisting of a line through f =1, which 
makes an angle Tr/2 (/> with the positive -axis, described from 
oo to 1, the -axis from 1 to 1, and this path reversed, we have 



nfl 



99,100] EXPRESSION FOR G n (z) 271 

" 



where in the latter integral 1 and f-fl have the amplitudes 
corresponding to the first description of the line. Hence 



so that 




Fio. 82. 



Now let f l = e X, so that X is real and positive; then 

Thus 



Accordingly, if R(TI + J)>0, this formula holds for all values 
of z such that 7r/2<<< 37T/2. 

100. Asymptotic Expansions of the Bessel Functions. In 
the formula obtained at the end of the previous section, let 
z\e~ i * = g, so that ( is real and positive ; then 

V-l niri .'{ 7T\ / 

(^r^^'H^-' 

provided K(7i + |)>0, - 7r/2 < < 37r/2. Hence 



272 



FUNCTIONS OF A COMPLEX VARIABLE [OH. xv 



where the integral is taken along a line making an angle y with 
the positive real axis such that ?r/2 < r\ < ?r/2. Thus 



Now let g=ue ir> , so that u is real and positive; also let 
u) = \fs(u) + ix(u), where \Js(u) and \(u) are real functions and 



Then 



x()= 



where 0<fl<l, 0<0'<1. Therefore 



^ 



Hence 



But 



2 , where = 



Now let amp ( 1 + ^ j = r ; 



then, if cos(>/ ^)=^0, as w increases from to oo, 7r<V<7r. 
Hence, if n = OL+i/3, and if s>a J, 



/ ie^t&V 1 --' 
\ 22 7 = 



100] ASYMPTOTIC EXPANSIONS 273 

Thus F^)!^, 

where M= ^"" -' 
Therefore 



It follows that 



1 



1 22 ;* 

But >; can always be chosen so that M is finite ; therefore, by 
sufficiently increasing \z\, | R, ] can be made arbitrarily small. 
Hence the series is asymptotic. 

Note 1. The expansion can be written 



(47i 2 -l 2 )(47i 2 -3 2 )(47i 2 -5 2 )(47i 2 -7 2 ) 



f (W-l 2 )(4? 
T" 2!(8^) 



'* '" 



2. Since G_ n (z) = e inn G n (z), the expansion also holds 
whenR(n+J)>0. 

Asymptotic expansion ofj u (z). Again, since J n (ze iir ) = e i?lir J 7l (z)> 
we can write 7riJ n (z) = G n (z) - e i ' i ' r G n (ze i ' r ). 

Thus, if 7r/2<^0<C7r/2, the asymptotic expansion for J n (z) 
is given by 



__ 

" 



2!(8z) 2 
(4n-l s )(4w f -3 2 )(4n*-5 2 )(4n*-7 s ) 



2 /(47i 2 -! 2 ) 

" 



/ 2 /(47i 2 
"V(7rz)l l! 



8z 3!(8z) 3 



M.F. 



274 FUNCTIONS OF A COMPLEX VARIABLE [OH. xv 

Also, since 3 n (z) = e in "J n (ze- i7r ), it follows that, when 
the expansion is 



" 12 (4tt 2 -l)(4n*-3)(4n-6) 
82 3!(8 2 js~ 



. 



Xsm 



COROLLARY. The difference between two consecutive zeros of 
J n (z) tends to the limit TT as z tends to infinity. 

Example. Prove 



where Jc is positive, and the quantities m g are the zeros of the function J (?w) 
regarded as a function of in. 

Since G (wa)J '(^)- J (na)G / (a)= , (cf. p. 241) 

1 1 



m s aJ l (m s a)' 



Now f5$ 

Jc 



FIG. 83. 



where C (Fig. 83) denotes a closed curve which crosses the .r-axis at the 
origin and at an infinitely distant point between two zeros of 3$(za\ and the 
summation extends to all positive values of m,. Therefore, since J (za) is an 
even function of 2, 



t 

J c 



But 

- 

Hence Lim 



=7r lf 



f J (ar)a-^flte. 



100] EXAMPLES XV 275 

EXAMPLES XV. 

1. If K(/3)>0, R(y-j8)>0, shew that 

II p- l (l-W-*- l (l-ztr*dt-*(fr y-)F(a, ft y, 0). 
Use this formula to prove Gauss's Theorem. 

2. If m is a positive integer, shew that 



deduce that 



3. If wi is a positive integer, shew that 



[Use Ex. 1, 97.] 

4. Use Ex. 1, 97, to prove that, if m and n are integers such that 
n ^ 0, m ^ - 7i, 



where C is a closed curve enclosing 

5. Establish the formulae : 

(i) P^iW^^PT^+^ 



(iii) (7i- 

[Apply the method of partial integration to the definite integral form 
for P m (4] 

6. Shew that the formulae of the previous example also hold for Q n w (z). 

7. If | z | < 1, shew that 



i 



+ 2\ r /l\ 
) L \S) 



2 

according as 1(2)^0. 

[Use Exs. VIII. 20, and Ex. 2, 62.] 

8. Prove that P n (^) = - tanmr{Q n (s)-Q_ n _i(z)K 



276 FUNCTIONS OF A COMPLEX VARIABLE [OH. xv 

9. Prove that 



[Use the second expression given in 97 for 
10. Prove that 



[Use Exs. 9, 2, 97.] 
11. Shew that 



(i) P M -( - z) = e^'^P n m (z) - - sin (n + m)7r- l Q B w (*), 

7T 



according as 

12. Shew that, if \z\>I, 



2" +1 cosmr r 



3 1\ 
' 2' ?/ 
.on 



, , 

[Use Ex. 2, 97.] 
13. Shew that, if |,|<1, 



2 



according as I (z} < 0. 
14. Shew that 



15. If R(w + *) > 0, shew that 



MISCELLANEOUS EXAMPLES. 

1. Shew that 



^ 
and give a geometrical interpretation of this equation. 

2. If n is a positive integer, prove that 

(i) z 2n - a? n = (z 2 - a*) (z* - 2az cos - + a 2 . . . - 2az cos 



( 
z 2 - 



3. Prove that, if the points z lt z 2 , 2 3 , are the vertices of an equilateral 

triangle, *? + Z* + Z 3 2 = Zfa + V 3 + Zfr . 

4. If 15 2 , z 3 , are the vertices of an isosceles triangle, right-angled 
at the vertex z 2 , prove that 



5. If (! - 3 2 )(V - 2 ') = ( 2 - 3 )(V - 3 ') = (Z 3 - ZjXV - /), 

shew that the triangles whose vertices are 2 1} 2 2 > z zt an( ^ ^i'* V? ^3'? are 
lateral. 

6. Similar triangles QRL, EPM, PQN, are described on the sides of the 
triangle PQR. Shew that the centroids of triangles PQR and LMN are 
coincident. 

7. If !, 2 , 3 , and b l9 6 2 , 6 3 , are the vertices of two triangles which are 
directly similar, shew that any three points which divide the line joining 
the pairs of points a lt b ; 2 ^2 '> a ^ ^3 5 i n tne same ratio, form a third 
similar triangle. 

8. If the lines joining z 2 and z 3 , z 3 and 2 15 z 1 and 2 , are divided in the 
same ratio r at z/, z 2 ', z 3 ', respectively, and if the triangles whose vertices are 
z u Z 2t z si an( i z ii z zi z s'i are similar, shew that either r= 1 or else both triangles 
are equilateral. 

9. Let ABCD be a parallelogram of which AC is a diagonal, and let 
ABX, DCY, ACZ, be similar triangles. Prove that triangle XYZ is similar 
to each of them. 

10. OCAD, OEBF, are circles, where O, A, B, C, D, E, F, are the points 
(0, 0), (2, 0), (6, 0), (1, 1), (1, -1), (3, 3), (3, -3), respectively. If 
w = *J{(I -z)(4-z)}, and if w=2 when z=0, find the values of w at A when 
z moves from O to A (i) along OCA, (ii) along ODA ; find also the values of 
w at B when z moves from O to B (i) along OEB, (ii) along OFB. 

Ans. -2*2 



278 FUNCTIONS OF A COMPLEX VARIABLE 

11. Shew that the equation w=\(z+^r^\ where z = re ie , determines a 
transformation which carries over circles, r= constant, and straight lines, 
0= constant, into confocal ellipses and hyperbolas respectively. Sketch the 
system of confocals. If P is any point within the circle || = 1, shew that 
there is a point Q outside that circle which is carried over into the same 
point of the w-plane as P is transformed into. 

12. If w=a(z c)/(z + c), where a and c are real and positive, shew that 
the interior of the circle | z \ = c in the z- plane corresponds to that half of the 
w-plane which lies to the left of the imaginary axis. 

13. If w=l/z, and if the point z describes that part of the line 4 t y = 3(#- 2) 
which lies in the first quadrant, find the path described by the point w. 
Shew on the same diagram the path described by w when z describes that 
part of the line 4y + 3(.#-2) = which lies in the fourth quadrant. Indicate 
in each case the direction of motion. 

Ans. Those parts of the circles 6u 2 + 6v 2 =3u4:V which lie in the fourth 
and first quadrants respectively. 

14. Shew that the transformation w=4/(z + l) 2 transforms the circle 
|^| = 1 into the parabola # 2 =4(1 u\ and that the interior of the circle 
corresponds to the exterior of the parabola. 

15. Shew that all the roots of 2 5 +2,s 2 + 2 + 3 = are in absolute value less 
than 1-6. 

[Cf. the proof of the Theorem of 10.] 

16. If a and b are real and positive, shew that the equation z* p + az + b=Q 
has 2p roots to the right, and 2p to the left, of the imaginary axis. If b is 
negative, shew that 2p+l roots lie to the right, and 2p-l to the left, of 
the imaginary axis. 

17. If a and b are real, shew that the equation z* p - l + az + b=Q has 2jo or 
2jp - 1 roots to the right of the y-axis, according as b is positive or negative. 

18. Prove that : (i) Lim (sec z - tan z) = ; 

cos 



(ii) Lim -^. - = *; (iii) Lim ^ - -=f. 

' z-^i sm-rrz z ^i simrz 

19. Shew that 

sin 2# i si 



20. If z tends to infinity along a straight line through the origin, shew 
that Lim tan 2= i, according as the line lies above or below the real axis. 

z >-oo 

21. If w=coshzj shew that the whole w-plane corresponds to any strip of 
the 2-plane of breadth TT bounded by lines parallel to the #-axis. Also shew 
that, to the lines x= constant, y= constant, correspond the confocal ellipses 
and hyperbolas, %2 ^ _ ^2^2^ 

- 2 ~ ' cos 2 sin 2 ~" 



MISCELLANEOUS EXAMPLES 279 

22. If w=\og{(z-a)/(z &)}, shew that the lines u= constant correspond 
to a coaxal system of circles whose limiting points are a and 6, whil6 the 
lines v= constant correspond to the orthogonal system. 

23. If z=ctanh(7ri0), shew that the lines u=u correspond to the coaxal 



and the lines v = v Q to the orthogonal system of coaxal circles. 

24. If the sequence z l1 z^ z 3 , ... , is convergent, shew that the sequence 

Zl+Z 2 Z 1 + 2 + Z 3 

15 ~T~* ~~3 '-' 

converges to the same limit. 

25. If the sequences z l , z 2t z 3 , ... , and z, z 2 ', 2 3 ', ... , converge to the limits 
z and / respectively, shew that the sequence i^, w 8 , w s , ... , where 



converges to the limit zz'. 

26. Integrate e a2 /(l+ e 2 ), where 0<a<l, round the rectangle whose sides 
are x= E, y=0, y=2?r, and shew that 

r e ax dx IT 

27. Prove that 

fe. 



|loglr,H-l<r<l, 

log ^ , if / < 1 or r > 1. 

Deduce that [log(cos 0) dO=j^ log (sin (9)rf(9=|logi. 

[Integrate -^A _| \ Z_ M_\ __J round the contour of Fig. 33, and 

put o?= tan #.] 

28. If - 2 < a < 2, prove that 



Deduce that, if - 2 < a < 2, 

sin 20(tan ff^dB^j sin 
[Integrate a f 2 round the contour of Fig. 37.] 



280 FUNCTIONS OF A COMPLEX VARIABLE 



29. By integrating <>g and og * where r ftnd g ftre real 

r iz r+iz 

and positive, prove that 



30. By integrating logf 1-f t*J ^ r 2 and logfl+i'-j 2 g .,, where r and 
are real and positive, prove that 



31. If a, c, and m are real quantities such that m g 0, c> 0, shew that 
sinm(x-a) dx v /, - mc 

~ 



ii p 

x-a 

32. Shew that, if a and b are real, and m^tt^O, 



/" sinm(x a)siun(x b) j _ sinn(a b) 
-- _ - QjX - 7T - ; - 
-< x-a x b a-b 



33. Prove that 



34. If ^ r < 1, shew that 

tf<9 27T 



'o l-Srcos^ + r 2 1-r 2 

/*2ir ,7/3 

35. Shew that J o y ^= 2?r or 0, according as | a \ < 1 or | a \ > 1 . 

36. Shew that, if | a n \ ^ 1 for all values of n, the equation 



cannot have a root whose modulus is less than ^. Also shew that the only 
case in which it can have a root z=*e ie is when a n = e~* ne , (w = l, 2, 3, ...). 

37. Shew that, if 



38. If | z | < 1, shew that 



39. Shew that 

,.. T . ,_ . ,, > , x .. x T . 1 cos(l 

(i) Lim (l+cos7r2)/tan 2 7T2}=J ; (n) Lim 



2 >0 



MISCELLANEOUS EXAMPLES 281 

40. Shew that, if | 2 | < 1 or | z ; > 1, the series 



_n ,. n 1 



has the sum s/{(s- 1)(2 2 

41. Prove that (i) | cos 2 1 ^ cosh 1 2 1 , (ii) \sinz |^sinh \z\. 
[Use the Taylor's Series for cos z and sin 2.] 

42. If |s | >1, shew that 



z+I 2 2 +l 
43. Shew that the series 



- -- 

2! 3! 

is convergent if E(s)>0, divergent if E(s)<0. 
44. Shew that the series 



is convergent for all values of z except 0, 1, - 2, - 3, ____ 

45. Shew that, for points interior to the circle 3.r 2 + 3y 2 + 2.?; - 1 = 0, 



1-* ' (l-zf (l-z) 3 ' "1-32 
46. Prove that, if | z \ < 1, and the principal value of tan-^ is taken, 



47. If \z\ <1, shew that 

22 3*3 



48. If . is neither zero nor a multiple of 2;r, shew that 

cosh 2- cos a. * f_ 2 2 ) 

1 +: 



49. Shew that 

sin* / 4_._.W, 4 . 2 \ /, 4 . 



50. Shew that 
(i) 



51. Shew that the series 

1_J_ + 1 J , 

~ 



1 + 22 2 + 2 3~3+l 

represents a meromorphic function with simple poles at the points 1, -2, 
-3,.... 



282 FUNCTIONS OF A COMPLEX VARIABLE 

52. If a is positive, shew that 

f coscu; , Tre~ a f 

I (H^^TTi* 

53. Shew that 



,, ,.v sin7r(2 + c) z + c-^r, (, z \ - 

54. Prove that (i) - . v = -- II. I 1- - }*i 

v ' SIRTTC c _ \ ?i-c/ 

,... sin 2 ?!-.? " /, ^ 2 

(ll) 1 -- r-s - = II 1 13 

- 



" ( . 4z* ~\ sin 3z 

55. Shew that II i 1 - 7 

-oo I (?nr + zy) smz 

56. Shew that ( " e- 2cose cos(^sin #)d<9 = |- f 

^0 2i J o x 

W Shew that V X ^ sinh (77^2) + sin ( 

' ~ 



co -r, ,, , " 1 

58. Prove that 2 



Si (?i+.r) 2 +3/ 2 y c 

59. Calculate the residues of the function (l+s 2 )-"- 1 , and shew that 
dx 1.3.5....(2ra-l) 



60. Shew that J x L = ^' 

61. Shew that, if m ^0, a > 0, 

/oo ^j a / Q\ 

I ^ gciajasj ^-g-( m+'-}' 

62. If - 1< E(0) < 3, shew that 

/" 

Jo 



o 

63. If ft is a positive integer, shew that 

ose cos (^(9 - sin 

64. If r\<l, shew that 



65. Shew that I 7 

Jc (z 



TTl 



( 

where C denotes the circumference of the circle ^ 2 +y 2 -2^-2y=0 described 
positively. 

66. If n is a positive integer, prove that 

P(cos fl-pgj {cos + j-^ .(- 2)9 



MISCELLANEOUS EXAMPLES 283 



[Expand both sides of the equation 

(i -sf cos tf+^-(i-f^a 

in powers of , and equate the coefficients of M .] 

67. If n is zero or a positive integer, shew that 

(i) P,n +1 (o)=o, (ii) P 2 ^o)=(-ir 1 

68. Shew that 



[Expand both sides of the equation 



and equate the coefficients of { 2n .J 

69. OB is one diagonal of a square OABC which has the side OA on the 
.r-axis and the side OC on the y-axis ; through D(2a, 2a), the mid-point of 
OB, lines are drawn parallel to OA and OC so as to divide OABC into four 
equal squares with sides of length 2a. If w is given by the series 

16 - 1 . 2m-lirx . 2n 



pr,ove that 

(i) w=0 along each side of the four squares ; 

(ii) w\ within each of the two squares about the diagonal ODB ; 
(iii) w= - 1 within each of the squares about the diagonal ADC. 

70. Integrate (1 - e~*)/z round the contour consisting of the positive x and 
?/-axes and a quadrant of an infinite circle, and shew that 

(i) 



71. If b and r are positive, and a is real, prove that 



(ii) 



72. Shew that, if o.>0, m>0, -Kr<l, 

x sin 2our TT 

-dx= f : 



sn GO? 



(io f-^ 

Jo m 2 + x*l- 2r cos 2out' + r 2 2 ( 1 + r) (e*" m - rV 

\ * / \ / 

flntegrate (i^ 



284 FUNCTIONS OF A COMPLEX VARIABLE 

73. Shew that the root of the equation z = +v:e z which has the value 
when 10 is given by n 



provided \iv\<\e~^~ l |. 

74. If z=+esinz, shew that, for small values of e, 

/\ / e ! & 3 2 

W ^=C+ r jSmf+2T 

e 

(ii) sin z = sin + ^ si 



sn cos + sn cos 



75. If z = (+wz m+1 , where =0, and if that root of the equation is taken 
which has the value f when w=0, shew that 



provided | w | < | m w (wi + i)--if- | . 

76. If n is a positive integer, shew that 
(i) P n '(^) = (2w 
(ii) P/(^) = (2?i 



77. If n is a positive integer, shew that the n zeros of ~P n (z) are all real 
and lie between 1. 

[Apply Rolle's Theorem to (# 2 - l) n and its derivatives.] 



78. Shew that, if n is zero or a positive integer, and if E(f)>0, 
T (cosh 2t-zfk'P n (z)dz 






79. If | r | < 1, shew that 



n 9 cos 20 , cos 3-^ 
(i) r cos 9 - r 2 - + r 3 - -- . . . = i log (1 + 2r cos + r 2 ), 

A O 



r n _ 



... 

where the principal value of the inverse tangent is taken. 
80. Prove that, if < < TT, 



81, Prove that 

(i) 

= -i log{4(cos 0-cosa) 2 }, 

(ii) cos cos a - i cos 2^ cos 2a. + J cos 3# cos 3a. - ... 

= Jlog{4(cos + cosa.) 2 }, 

unless one of the quantities 6 -a. and 6+a, is an even multiple of TT in case 
(i) or an odd multiple of TT in case (ii). 



MISCELLANEOUS EXAMPLES 285 

82. Shew that, if ^ 6 ^ 2:r, 

(i) cos (9+^ cos2<9+^cos3(9+...= 1 3 2(3(9 2 - 

(ii) sm<9+j3sin2^ + ^3s 

83. If ^ ^ TT, shew that 



-fsin 2 6log(4sin 2 0), 
, x sin 40 sin 66 sin 80 . n/1 , _ m . 9/3 

(n) Y^+-JT^ + ~3~T + ''' = * m ^~ ^ 

- sin cos 6 log (4 sin 2 0). 

84. If n is a positive integer, and if 1 2 1 < 1, shew that 



Deduce that 



_ 



85. If ^ 6> ^ TT, shew that 

TT ,. x sin 6 sin 3^ sin 



86. If - 7T/2 ^ ^ ^ 7T/2, shew that 

(9 2 \ . sin ZQ sin 5(9 



87. If - 7T/2 ^ ^ ^ 7T/2, shew that 

cos 30 cos 50 cos 76 TT 



oo a 'I2 

88. Shew that the series 

=i J 

represents a continuous function in the part of the 2>plane for which 
100 = 0, and that the function is holomorphic at all points below the real 

axis. 



Ort -r, 1 2 / COS27T^ COS37TJI' \ 

89. Prove that ;y = - + -^( COSTTJ; -- ,^5 + ^ -- ... 

o 7T \ A" v J 

represents a series of equal and similar parabolic arcs standing in contact 
along the a'-axis. 



90. Prove that J\ - - .* 70X = 21 coth ira coth 7r6. 

m^-oo ,,= C/ 2 

91. If -KR(a)<l, shew that 

rsinh ajc dx 
-- 
cosh.r a? 



286 FUNCTIONS OF A COMPLEX VARIABLE 

Deduce that, if A. is real, 

rsin Xx dx . / , vrA\ 

r -- = 2 tan- 1 1 tanh I 
cosh# x \ 4 / 

92. Prove that Lim ( 1 + \ + \ + . . . +^- T - \ log M) = log|2 + |y. 

,t_> C) o \ o o zn \.& / 

93. Shew that 



94. Shew that 



[Use the identity ( -!)-- 2(0"* - 1 )-' = ( 2 + 1)- 1 .] 



[Shew that e- a - ; e- 2a +... + (- 1 ) n ~ 1 6- na = ^. f ^^ z e ~ azdz > where C is tlie 
contour of Fig. 58, and use Ex. 94] 

96. Shewthat (i) cot=l-B 1 -B 2 -B 3 -... , 



2!4 4!6 6! 



97. Prove that (i) l-^^-i, (ii) 

w =2 l\ */ J H^Afr+l/ 3 

98. If U(n)> 0, shew that 

(cos ^) n ~ 1 cos (a tan ^) cos (w + 1) dO 

= P(cos 1 ^) n - 1 sin (a tan <9)si 

^o 



[Use Exs. VIII. 6.] 

99. Shewthat *() 

100. Shew that, if m is a positive integer, 



101. Shew that 

(i) V^)= 2 , (ii) ^(0) = 2 , (iii) V^(-i) = - 



102. Prove, by using the equation 



MISCELLANEOUS EXAMPLES 287 

that, if R(c)> - 1, 



103. Shew that, if B(0)> - 1 and R(fc)>l, 

e -tz t n~\ 



) H (0 + 2)" (0 + 3) n r(w)Jo e'- 

104. If E(a)>0, shew that 

105. Shew that, if R(0)> - 1, 



106. Shew that, if R (71) > 1 , 

cos < n = ~ cos ;<) cos 



If 7i is zero or a positive integer, prove, by considering the cases n even 
and n odd separately, that 



Jo c -~v-T-/x"~-r/ -r ..(n + k\(n + k n \ (n+k 

^M(-i- l )~\-^- n 

Deduce that, if n is zero or a positive integer, 

^ l"^^./7f\/ / \* 71 W I 

7T Jo 



rTTi ^57Ti 

V^cfo, where z Q = ae 4 , =ae 4 , and the path of integra- 


tion is a semi-circle of centre the origin and radius a described positively. 
Also find the values of the integrals which have Z Q as initial point, and 
whose paths are : (i) a complete circumference of the circle ; (ii) two cir- 
cumferences ; (iii) three circumferences. What is the shortest non-zero 
path from ZQ along the circumference which makes the integral zero ? 

4 / ^''X A / ? -'\ 

Ans. -fV&a*; (i) -|a*(l+?J, (ii) -fHe 3 +e 3 ), (iii) 0; three- 
fourths of a circumference. 

108. Prove that f ;/RT _|L___ =F (,, |). 

109. Shew that all elliptic integrals /E(^, JX.)dx, where R(A', y) is 
rational in x and ?/, and X is a cubic in x with no repeated factors, can be 
expressed in terms of integrals of the three types 

f dy f ydy f dy 

J VC--)' J ^--- J -' 



288 FUNCTIONS OF A COMPLEX VARIABLE 

110. Establish the identity 



where the product is taken for all integral values of A and /x from to n t 
with the restriction 

111. Shew that 



W 



(n+l)* 



112. Shew that 

113. Shew that 



where pM 

114. Prove that 



du 



115. The function <(p(u} has a real period 2^ and an imaginary period 2o> 2 , 

where w 2 --= ^log ( - ), a and 6 being real and positive, and such that a<b. 

TT \aj 

Shew that, if 0=fj(f^llogi ), the annulus in the ^-plane bounded by the 

\ 7T Oil 

circles |f| = a and |f| = 6 and a barrier along the positive real axis, corre- 
sponds to the entire 2-plane. Shew also that only one point of the annulus 
corresponds to each point of the 2-plane. 

116. Prove that 



sn 



117. Shew that 



/ \ 
- sn(- ,) = 



Lim 



-*o M* 8 

118. If six of the nine points in which the cubic y 2 = 4x 3 -g 2 x-g 3 is cut 
by a second cubic lie on a conic, shew that the other three points lie on a 
straight line. 

119. If a conic passes through four fixed points on the cubic 

y*=^-g&-gv 

shew that the straight line joining the two variable points of intersection 
passes through a fixed point on the cubic. 



MISCELLANEOUS EXAMPLES 289 

120. Solve the equation w" + az 2 iv = Q. 



121. Solve the equation vf" + -Izw' + w = 0. 

Ans. w i = l -^ + -^T zQ - ' 9'; 

1 3 , 3.9 3.9.15 



2 10 +..., 

1 2_A*5 5.11 8 5.11.17 n 
W *~2\ Z ~5~! 8! Z 11! 

122. If n is a positive integer, shew that all the zeros of P n (z) are 
simple zeros. 

[By differentiating Legendre's Equation it can be shewn that if P(.z) has 

a zero of the second order, ~ P n (z) = for all positive integral values of .] 

123. Find ttiat integral of the equation 



which has the value unity when z = 0. Ans. 



Find regular integrals in the neighbourhood of 2 = for the equations of 
Examples 124-128. 



124. 42%" + 4zw'- 
Ans. 



2 1\ 23 /2 2 1 

^ =wz+z 'l-^r 

125. 



oo -n+4 

Ans. wz^e* w 



126. z*(z + l)w"-z*w' + %(Zz+l)w = 0. Ans. w^z, w 2 

127. 22 2 (2-2)w // -2(4-2)w' + (3-0)w = 0. Ans. w 1= z^ w 2 =(z- 

128. 2 2 (l-2)^y" + 2(5^-4)w' + (6-92)^<;=0. J?i5. w^^ 3 , w> 2 = ^ log + z 2 . 

129. If n is zero or a positive integer, shew that 



dz n+ 
130. Shew that, for all values of 



M.F. 



290 FUNCTIONS OF A COMPLEX VARIABLE 

131. Shew that, for all values of n, 

(i) (1 -z^r n (z}=nP n ^(z)-nzP n (z), 

(ii) (l-^)PLl 

(iii) (1 -z*)Q n (z 

(iv) (1 -^QL.M + nQnW -nzq n 

132. If n is zero or a positive integer, shew that 



where W n -i(z) is a polynomial of degree n l. 

[in Ex. 1, 90, write <*,=! ^ df-| ^ ? 

133. With the notation of Example 132, shew that 



[Substitute the expression obtained for Q n () in Ex. 132 in Legendre's 
Equation, put W w _ 1 (2) = a 1 P n _ 1 (^)-f-a 3 P n _ 3 (^)+... , and use Example 76.] 

134. Shew that 



deduce that, if m and ^ are positive integers, both odd or both even, 



while if m is an even and n an odd integer, 

r p m (z)p w (s)^=(-i) 

j 



-1 ,\ 2 

27V 2~\ 
[Cf. proof of Exs. XIV. 5, and use Ex. 131.] 

135. If m and n are integers such that w^O, m^w, shew that 

Pn~ m O?)= ^gn^; 

136. Prove that, if E(n) > R(m) > - 1, 



and deduce the results : 



9 ^n 

(i) Jn(z) = ^ ~ M2 ) M ^ COSigMrfM > where 



(ii) _ = J (^sin 6)sin 

2 JO 

[Expand J TO (2w) in powers of w, and integrate.] 



MISCELLANEOUS EXAMPLES 291 



137. Solve the equation zi&" + (n + l)u/-w = 0. 

An*. w^s-Sj^V:), 

138. Shew that, if n is an odd positive integer, 



[Use the formula 2.J n (z) = J n _i(.r) + J n+ i(4] 
139. Tf is an integer, shew that 



[In EXS. XIV., 14, put f=e ifl ,and = <j>-irl2.] 
140. Prove that ,? 2 = 2 (2rc) 2 J 2 (4 

n=l 

[Differentiate the equation e^ : ^- 1 /0= 2Jt*){" w ^h regard to ^, multiply 
by , differentiate again, and put = 1 .] 

141 . Prove that (i) cos x = J (x) - 2 J 2 (a?) + 2 J 4 (.r) - . . . ; 

(ii) sin x = 2.1, (.?;) - 2 J 3 (.r) + 2 J, (.r) 
[In Exs. XIV., 14, put =i.] 

142. Shew that (!) J) 



143. Shew that 



144. Shew that, if p is a positive integer, 




[Use Exs. XIV., 11.] 

M.F. T2 



292 FUNCTIONS OF A COMPLEX VARIABLE 

145. If n is a positive integer, shew that 

' 



(iii) r 
146. Prove that 

T MJ M= 



[Shew that the coefficient of f - j in the product i 



and apply Gauss's Theorem.] 

J47 Shew that, if n is zero or a positive integer, 

2 r* 

- / J n (2z cos 0) cos (M) o?</) = J ll+t M J )( _ A . (:). 

7T JO Y~ ~2~ 

[Expand J n (2iCos<^) in powers of cos<, and use Examples 106 and 146.] 
148. If x and u are real, prove that 



[Use the relation - "!'' '- = uS 






i prove that 



149. If .r is real, and 



[Use Exs. XIV., II.] 

150. If and 6 are positive constants, prove that 




.r cos 



/ * 30 

/ 



[Put J O (&F) = ~ | cos (6. 
of integration.] 

151. If Ii(b ?)>0, shew that 

Jo ^ 

[Put J (a.r) = - I cos (a.r cos 




(Exs. XIV., 18), and change the order 



and change the order of integration; 

or, expand J Q (CLT) in powers of .T, and integrate term by term (cf. Bromwich, 
Infinite Serf ex, 176, B).] 



MISCELLANEOUS EXAMPLES 293 

152. Shew that, if R(2?i + 1)>0, and E(6 ia)>0, 



[For (i) use the substitution given in Exs. XIV., 18, for J n (#.r), and change 
the order of integration ; after the first integration expand (6 + ia cos c^)- 2 "- 1 
in powers of cos <, and integrate again ; or, expand J(o#) in powers of #, 
and integrate term by term. For (ii), differentiate (i) with regard to b.] 



153. Prove that - / * e** cos * cos (y sin <) d<j> = J 

7TJO 



[Expand cos (y sin (/>) in powers of sin 0, and apply 99, Cor., Example 
145, (iii), and Taylor's Theorem.] 



INDEX. 



The numbers refer to the pages. 



Abel's test for convergence of series, 80. 

theorem on continuity of series, 125. 
Addition of complex numbers, 1, 3. 
Amplitude, 2, 4. 

of a function, variation of, 11-17. 

principal value of, 2, 4. 
Argand diagram, 2. 
Argument, 2. 
Asymptotic expansions, 136. 

(See under Bessel, Euler, Gamma 
function.) 

Bernoulli numbers, 132. 
Bessel function, 236. 
Bessel function G n (z), 240. 

addition theorem for, 254. 

as a contour integral, 267. 

asymptotic expansion of, 271. 

in terms of Bessel functions of first 
and second kinds, 240. 

recurrence formulae for, 241. 
Bessel function of first kind, 237. 

addition theorem for, 254. 

as a contour integral, 267, 268, 270. 

as a function of its order, 239. 

asymptotic expansions of, 273, 274. 

recurrence formulae for, 239. 

zeros of, 241, 274. 

Bessel function of second kind, 238, 239. 
Bessel functions, relations between, 241. 
Bessel's equation, 236, 266. 
Beta function, 144, 145. 
Binomial theorem, 90. 
Branch of function, 13. 
Branch point, 14, 39. 

of an integral, 257. 

Cauchy's integral theorem, 51, 54. 

residue theorem, 57. 
Circular functions, 33, 83, 90. 
Coefficients, undetermined, 9G. 



Collinearity of points on cubic, 197. 
Complex numbers, 1. 

geometrical representation of, 1 . 

operations with, 1-5. 
Complex variable, 7. 

function of a, 7. 

path of variation of a, 7. 
Conformal representation, 37. 
Congruent points, 179, 180. 
Conjugate numbers, 1, 2. 
Connected region, 30. 
Continuation, analytical, 122, 208. 

of hypergeometric function, 153, 156, 
249. 

of integral of diif. equation, 213. 

theorems on, 123, 124. 
Continuity, 23, 24. 

Abel's theorem on, 125. 

of series, 92. 

uniform, 24-26. 
Convergence of infinite product, 107, 108. 

unconditional, 107. 
Convergence of sequence, 42. 

uniform, 42. 
Convergence of series, 76. 

absolute, 76, 78. 

circle of, 80. 

of a double series, 78. 

of power series. 80, 82, 95. 

radius of, 80. 

ratio tests for, 77. 

uniform, 92. 
Coordinates, polar, 2, 29. 

rectangular, 2. 
Cross-cut, 30. 

Derivative, 26, 28. 

of function of a function, 30. 
of holomorphic function, 28, 70. 
of inverse function, 30. 
partial, 31, 70. 



INDEX 



295 



The numbers refer to the pages. 



Determinant of fundamental system, 
216, 223. 

index of, 224. 

Differential equation, homogeneous 
linear, 209. 

coefficients of, 210. 

construction of, 216. 

domain of ordinary point of, 210. 

dominant equation, 210. 

Frobenius' method of solution, 225. 

fundamental equation, 220. 

fundamental system, 215, 257. 

indicial equation, 225. 

integrals of, 210. 

of the first order, 210. 

of the second order, 210. 

ordinary point of, 210. 

singularity of, 210. 

solutions of, 210. 
Differentiation, 26, 28, 29. 

of series, 93. 

under integral sign, 44, 69, 138. 
Discontinuity, removable, 23. 
Division of complex numbers, 1, 4. 
Domain of a point, 38, 210. 

Elements of a function, 208. 
Elliptic function, 180. 

order of, 181, 182. 

poles of, 180 to 183. 

zeros of, 182. 

(See under Jacobian and Weier- 

strassian functions.) 
Elliptic integrals, 169. 

reduction of, 170-173. 

transformation of, 170-174. 

(See also Legendre's and Weier- 

strass's elliptic integrals.) 
Equations, roots of, 16, 69. 
Euler's constant, 135. 

asymptotic expansion of, 134. 
Euler's definition of gamma function, 

141. 

Expansion, Lagrange's, 119. 
Expansion of functions in scries of 

fractions, 103, 105. 
Exponential function, 32, 90. 

Fourier series, 86. 

Frobenius' method of solving linear 

diff. equations, 225. 
indicial equation, 225. 
solutions free from logarithms, 228. 
uniform convergence of series with 
regard to index, 227. 



Fuchsian type, equations of, 243. 

sum of indices a constant, 243, 244. 
Function, analytic, 29, 208. 
conjugate, 31. 
continuous, 23. 
dominant, 210. 
doubly-periodic, 179. 
elements of a, 208. 
even, 33, 97. 

geometrical representation of a, 7, 10. 
holomorphic, 29, 52, 93. 
initial value of, 10. 
integral, 88. 
integrals of, 48. 
inverse, 30. 
limit of, 22. 

meromorphic, 39, 40, 89, 160. 
multiform or multiple-valued, 7, 161, 

209. 

odd, 33, 97. 

of a complex variable, 7. 
of a function, 24, 30, 49. 
of two complex variables, 69, 137. 
of two real variables, 26. 
periodic, 32, 86. 

periodic, of the second kind, 187. 
periodic, of the third kind, 189. 
rational, 89. 
rational integral, 88. 
region of existence of, 7. 
regular, 29. 
simply-periodic, 86. 
single-valued, 7, 209. 
transcendental integral, 88. 
uniform, 7, 209. 
uniform, classification of, 88. 
Fundamental equation, 220. 
Fundamental system of integrals, 215, 

257. 

associated with fundamental equa- 
tion, 220. 

in neighbourhood of singularity, 219. 
Fundamental theorem of algebra, 68, 69. 

Gamma function, 109, 139, 141. 

asymptotic expansion of, 146. 

duplication formula for, 145. 

Euler's definition of, 141. 

expression as a contour integral, 143. 

Gauss's definition of, 141. 

the derived function \f(2), 141. 
Gauss's differential equation, 228, 258. 

function 11(2), 141. 

sum, 117. 

theorem, 144. 



206 



FUNCTIONS OF A COMPLEX VARIABLE 



The numbers refer to the pages. 



Geometrical representation. (See under 
Complex numbers, Functions and 
Transformations. ) 

Green's theorem, 45. 

Gregory's series, 84. 

Harmonic functions, 31. 
Harmonics, cylindrical, 236. 

spherical, etc., 249. 
Hyperbolic functions, 33, 90. 
Hypergeometric equation, 228, 258. 

relations between integrals of, 249. 

the twenty-four integrals of, 247. 
Hypergeometric function, 77, 151, 229, 
246, 247. 

analytical continuation of, 153, 156, 
249. 

as a contour integral, 259. 
Hypergeometric series, 77, 78, 144. 

convergence of, 77. 

Identities, 83. 
Image of point, 9. 
Indented contour, 65. 
Indicial equation, 225. 

fundamental system, 227. 
Infinity, point at, 9. 

continuity at, 23. 

integral at, 51, 137, 139. 

integrals of diff. equ. at, 212, 213, 224. 

loop about, 162, 168. 

residue at, 58, 96. 

singularity at, 38, 39. 
Integrals, contour, 59, 97, 113. 

convergent, 136. 

curvilinear, 42. 

definite, 48. 

double, 69, 138. 

elliptic, 169. 

evaluation of definite, 59, 97, 113. 

finite moduli of definite, 50. 

Fresnel, 62. 

indefinite, 53. 

independent of paths, 52. 

limiting values of definite, 60, 63, 
113, 115. 

of holomorphic functions, 50-52. 

of meromorphic functions, 160. 

of multiform functions, 161. 

piincipal values of, 65. 

uniformly convergent, 137. 

with infinite paths, 51, 137, 139. 
Integrals of differential equation, 210. 

analytical continuation of, 213. 

at infinity, 212, 213. 



Integrals of differential equation, 

existence of, 210. 

fundamental system of, 215. 

in form of infinite series, 213. 

initial values of, 210. 

linearly independent, 215. 
Integrals of diff. equ. in form of definite 
integrals, 255, 266. 

branch points of, 257. 

fundamental system of, 257. 
Integrals of diff. equ. near a singularit} 7 , 
219. 

at infinity, 224. 

fundamental system of, 219. 

index of, 222. 

regular, 222. 

Integrand, infinite, 136, 139. 
Integration, change of order of, 69, 138. 

of series, 93. 

partial, 53. 

under integral sign, 69, 138. 
Invariants (see under Weierstrass). 
Inverse points, 9. 
Inverse sine function, 163. 
Inverse tangent function, 34, 84, 86. 
I(p) notation, 1. 

Jacobian elliptic functions, 167,182, 198. 

addition theorems for, 202. 

complementary modulus of, 166, 167, 
200. 

derivatives of, 200. 

diff. equ. of quarter periods of, 176, 
231. 

duplication formulae for, 204. 

Legendre's relation for, 175. 

moduli of, 167, 200. 

orders of, 182, 202. 

periods of, 167, 201, 202. 

poles of, 200, 202. 

relations between, 200. 

relations between periods of, 202. 

relation to Weierstrassian functions, 
201. 

residues at poles of, 202, 205. 

transition from Weierstrassian func- 
tion to, 198. 

zeros of, 200. 
Jacobi's imaginary transformation, 205. 

Lagrange's expansion, 119, 125. . 
Landen's transformation, 174. 
Laplace's equation, 31. 
Laurent's series, 84, 95. 

absolute convergence of, 85. 



INDEX 



297 



The numbers refer to the pages. 



Legendre functions, 249. 

of the first kind, 214, 235, 236. 

of the second kind, 235, 236. 

recurrence formulae for, 236, 252, 

289, 290. 
Legendre polynomials, 99, 214, 235. 

expression in series, 102, 103, 121. 

in definite integral forms, 100-102. 

integrals involving, 122. 

recurrence formulae for, 102, 124, 
129. 

Rodrigues' formula for, 120. 
Legendre's associated equation, 249, 

259. 
Legendre's associated functions, 250. 

as definite integrals, 260-263. 

relations between, 251, 262-265, 275, 

276. 
Legendre's complete elliptic integrals of 

the first and second kinds, 174. 
Legendre's equation, 213, 234. 

relation to Gauss's equation, 235. 
Legendre's first normal elliptic integral, 
163, 173. 

inversion of, 166, 201. 
Legendre's normal integrals, 173. 
Legendre's relation, 175, 188. 
Limit, 22. 

at infinity, 22. 

infinite, 23. 

of a sequence, 42. 

of function, geometrical illustration, 
22. 

of ratio of two functions, 30, 83. 

uniform convergency to a, 23. 
Liouville's theorem, 68. 
Logarithmic function, 34-36, 83, 161. 
Logarithmic transformation, 35. 
Loops, 145, 162, 164, 168. 

about point at infinity, 162, 168. 

notation for negative, 162. 

Mittag-Leffler's theorem, 105. 

Modulus, of complex number, 2, 3, 4. 
(See under Jacobian elliptic func- 
tions.) 

Multiplication of complex numbers, 1, 
4. 

Naperian logarithms, 34. 

Numbers, complex, imaginary, real, 

1,2. 
geometrical representation of, 1-6. 

Orthogonal systems, 32. 



Path of variation, 7, 10, 22. 
Period, of a function, 80, 179. 

parallelogram, 179. 

primitive, 86, 179. 
P-function, Biemann's, 244. 
Point at infinity, 9, 38, 39. 
Points, congruent, 179, 180. 
Points, critical, 38. 
Points of inflection on cubic, 197. 
Points, ordinary, 38, 210. 
Points, singular, 38. 
Pole, 38, 39, 67, 118. 

an isolated singularity, 39. 

at infinity, 38, 88. 

of order n, 38, 86. 

principal part at a, 86. 

simple, 38. 

Polynomials, 88. 
Power, the generalised, 36. 
Product, infinite, 107, 108. 

expression of function as, 108, 109. 

Quantities e, rj, positive, 23. 

Region, closed, 92. 

connected, 30. 

function holomorphic in, 53. 

multiply-connected, 30, 47, 58. 

of existence of function, 7. 

of uniform convergence, 92, 96. 

simply-connected, 30. 
Residue at a pole, 57, 58, 67, 96. 

at infinity, 58, 96. 
Riemann's P-function, 244. 

indices of, 245. 

in terms of hypergeometric functions, 

246. 

Rodrigues formula, 120. 
Root extraction, 1, 5, 36. 
Roots of equations, 4, 5, 10, 69. 

theorems on, 118, 119. 
R(p) notation, 1. 

Sequence, 42. 

Series, convergent, 76. 

multiplication of, 77, 82. 

power, 80, 82, 95, 125. 

uniformly convergent, 92. 
Sigma functions, 109. 

duplication formula for, 190. 

elliptic functions in terms of, 190. 

properties of, 189. 
Similar figures, 8, 37. 
Singularities, 38. 

at infinity, 38, 39, 88, 89, 106, 181. 



298 



FUNCTIONS OF A COMPLEX VARIABLE 



The numbers 

Singularities, 

essential, 39, 86, 89, 90, 106, 181. 

isolated, 38, 39. 

line of, 101. 

non-essential, 39. 

of a diff. equ., 210. 
Stirling's formula, 150. 
Sturm's theorem, 16. 
Subtraction of complex numbers, 1, 3. 
Summation of series by residues 

116. 

Summation of trigonometrical series 
126, 127. 

Tangent to a cubic, 197. 
Taylor's series, 82. 95. 

absolute convergence of, 83. 
Transformations, 7. 

bilinear, 8, 9. 

geometrical representation of, 8. 

linear, 7, 8. 

rational, 8. 

(See under Landen, Logarithmic.) 
Trigonometrical series, summation of, 
126. 

Uniformly convergent series, 92, 127. 
continuity of, 92. 
differentiation of, 93. 
integration of, 93. 
power series, 95. 
Weierstrass's M test for, 94. , 

Variable, complex, 7. 

independent, 7. 
Vectors, 2. 



refer to the pages. 

Weierstrassian elliptic function, 106, 

169, 180. 
addition of semi-period to argument, 

187. 

addition theorem, 185. 
diff. equation satisfied by, 183. 
duplication formula for, 186. 
elliptic functions in terms of, 191. 
geometric application of, 196. 
in terms of sigma functions, 190. 
invariants of, 184, 194. 
Legendre's relation for, 188. 
order of, 182. 

periods of, 169, 180, 195, 196. 
poles of, 181, 182. 
relation to Jacobian functions, 201. 
residue at pole of, 181. 
transition to Jacobian functions, 198. 
values when one period real and one 

purely imaginary, 194. 
zeros of first derivative of, 182, 184. 
Weierstrassian elliptic integral, 167, 

185, 195, 196. 
inversion of, 169, 185. 
Weierstrass's theorem, 108. 
Weierstrass. (See under Sigma and 
Zeta functions and Uniformly con- 
vergent series.) 
w-plane, 10. 

Zeros, 1,39,67, 118, 119. 

of order n, 39, 83. 

simple, 39. 
Zeta functions, Weierstrass's, 106. 

elliptic functions in terms of, 188. 

properties of, 187. 
z-plane, 2, 10. 



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